Review Article A Review on Unique Existence Theorems in ...downloads.hindawi.com/archive/2014/835394.pdf · areas of research in di erential geometry. Most of the work ontheRiemannian,semi-Riemannian,andLorentzianman-ifolds

Review ArticleA Review on Unique Existence Theorems in Lightlike Geometry

K L Duggal

Department of Mathematics and Statistics University of Windsor Windsor ON Canada N9B 3P4

Correspondence should be addressed to K L Duggal yq8uwindsorca

Received 10 March 2014 Accepted 27 May 2014 Published 7 July 2014

Academic Editor Yuji Kodama

Copyright copy 2014 K L Duggal This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This is a review paper of up-to-date research done on the existence of unique null curves screen distributions Levi-Civitaconnection symmetric Ricci tensor and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (inparticular Lorentzian) manifolds supported by examples and an extensive bibliography We also propose some open problems

1 Introduction

The theory of Riemannian and semi-Riemannian manifolds(119872 119892) and their submanifold is one of the most interestingareas of research in differential geometry Most of the workon the Riemannian semi-Riemannian and Lorentzian man-ifolds has been described in the standard books by Chen[1] Beem and Ehrlich [2] and OrsquoNeill [3] Bergerrsquos book [4]includes the major developments of Riemannian geometrysince 1950 covering the works of differential geometers ofthat time andmany cited therein In general an inner product119892 on a vector space V is of type (119903 ℓ 119898) where 119903 = dim119906 isin

V | 119892(119906 V) = 0 for all V isin V ℓ = supdim119882 | 119882 sub

V with 119892(119908119908) lt 0 for all nonzero 119908 isin 119882 and 119898 =

supdim119882 | 119882 sub V with 119892(119908119908) gt 0 for all nonzero 119908 isin

119882 Kupeli [5] called a manifold (119872 119892) of this type a singularsemi-Riemannian manifold if 119872 admits a Koszul derivativethat is 119892 is Lie parallel along the degenerate vector fieldson 119872 Based on this he studied the intrinsic geometry ofsuch degenerate manifolds On the other hand a degeneratesubmanifold (119872 119892) of a semi-Riemannian manifold (119872 119892)

may not be studied intrinsically since due to the degeneratetensor field 119892 on 119872 one cannot use in general the geometryof 119872 To overcome this difficulty Kupeli used the quotientspace 119879119872lowast = 119879119872Rad(119879119872) and the canonical projection119875 119879119872 rarr 119879119872lowast for the study of intrinsic geometry of 119872where Rad(119879119872) is its radical distribution

For a general study of extrinsic geometry of degeneratesubmanifolds (popularly known as lightlike submanifolds)of a semi-Riemannian manifold we refer to three books

[6ndash8] published in 1996 2007 and 2010 respectively Asubmanifold (119872 119892 119878(119879119872)) of a semi-Riemannian manifold(119872 119892) is called lightlike submanifold if it is a lightlikemanifold with respect to the degenerate metric 119892 inducedfrom 119892 and 119878(119879119872) is a nondegenerate screen distributionwhich is complementary of the radical distribution Rad(119879119872)

in 119879119872 that is

119879119872 = Rad (119879119872)oplusorth119878 (119879119872) (1)

where oplusorth is a symbol for orthogonal direct sum The tech-nique of using a nondegenerate 119878(119879119872) was first introducedby Bejancu [9] for null curves and then by Bejancu and Dug-gal [10] for hypersurfaces to study the induced geometry oflightlike submanifolds Unfortunately (i) the induced objectson119872 depend on 119878(119879119872)which in general is not uniqueThisraises the question of the existence of unique or canonicalnull curves and screen distributions in lightlike geometry(ii) The induced connection nabla on 119872 is not a unique metric(Levi-Civita) connection and depends on both the inducedmetric 119892 and the choice of a screen which creates a problemin justifying that the induced objects on 119872 are geometricallystable (iii) The induced Ricci tensor of119872 is not a symmetrictensor so in general it does not have a geometric or physicalmeaning similar to the Riemannian Ricci tensor and (iv)since the inverse of degenerate metric 119892 does not exist onefails to have well-defined concept of a scalar curvature bycontracting Ricci tensor At the time of the 1996 book [6]nothing much on the above anomalies was available In 2007I published a report with limited information available on

Hindawi Publishing CorporationGeometryVolume 2014 Article ID 835394 17 pageshttpdxdoiorg1011552014835394

2 Geometry

how to deal with this nonuniqueness problem for null curvesand hypersurfaces [11] Since then considerable further workhas been done on these issues in particular reference to alltypes of null curves and submanifolds which has providedstrong foundation for the lightlike geometry

The objective of this second report is to review up-to-dateresults on canonical or unique existence of all types of nullcurves and screen distributions and then find those lightlikesubmanifolds which also admit a uniquemetric connection asymmetric Ricci tensor andhow to recover the induced scalarcurvature subject to some reasonable geometric conditionsWe also propose open problems Our approach is to give briefinformation on themotivation for dealingwith each anomalychronological development of themain results and a sketch oftheir proofswith examples In order to include a large numberof results in one paper we provide a good bibliography withthe aim to encourage those wishing to pursue this subjectfurther More details on these and related works may be seeninBibliography of LightlikeGeometry prepared by Sahin [12]

2 Canonical or Unique NongeodesicNull Curves

Let119862 be a smooth curve immersed in an (119898+2)-dimensionalproper semi-Riemannian manifold (119872 = 119872119898+2

119902 119892) of a

constant index 119902 ge 1 By proper we mean that 119902 is nonzeroWith respect to a local coordinate neighborhoodU on 119862 anda parameter 119905 119862 is given by

119909119894

= 119909119894

(119905) 119894 isin 0 119898 + 1

rank (1198891199090

119905sdot sdot sdot 119889119909

119898+1

119905) = 1 forall119905 isin 119868

(2)

where 119868 is an open interval of a real line and we denote each119889119909119894119889119905 by 119889119909119894

119905 The nonzero tangent vector field onU is given

by 119889119905equiv (1198891199090

119905 119889119909119898+1

119905) equiv 120585

Suppose the curve 119862 is a null curve which preserves itscausal character Then all its tangent vectors are null Thus119862 is a null curve if and only if at each point 119909 of 119862 we have119892(120585 120585) = 0 The normal bundle of 119879119862 is given by

119879119862perp

= 119883 isin Γ (119879119872) 119892 (119883 120585) = 0

dim (119879119862perp

)119909= 119898 + 1

(3)

However null curves behave differently compared to thenonnull curves as follows

(1) 119879119862perp is also a null bundle subspace of 119879119872(2) 119879119862 cap 119879119862perp = 119879119862 rarr 119879119862 oplus 119879119862perp = 119879119872

Thus contrary to the case of nonnull curves since thenormal bundle 119879119862perp contains the tangent bundle 119879119862 of 119862the sum of these two bundles is not the whole of the tangentbundle 119879119872 In other words a vector of 119879

119909119872 cannot be

decomposed uniquely into a component tangent to 119862 and acomponent perpendicular to119862 Moreover since the length ofany arc of a null curve is zero arc-length parameter makes nosense for null curves For these reasons in general a Frenet

frame (constructed by Bejancu [9] in 1994) on a Lorentzianmanifold 119872 along a null curve 119862 depends on the choiceof a pseudoparameter on 119862 and a complementary (but notorthogonal) vector bundle 119878(119879119862

perp) to 119862 in 119879119862perp calling itsscreen distribution In the following we review how onecan generate a canonical or unique set of Frenet equationssubject to reasonable geometric conditions We discuss thisin two subsections of null curves in Lorentzian and semi-Riemannian manifolds (of index 119902 gt 1) respectively

21 Null Curves in Lorentzian Manifolds (119902 = 1) The mainidea (first used by Cartan [13] in 1937 followed by Bonnor[14] in 1969) is to choose minimum number of curvaturefunctions in the Frenet equations We need the following twogeometric conditions on a curve 119862(119901)

(a) 119862(119901) is nongeodesic with respect to a pseudo-arc-parameter 119901

(b) choose119862(119901) such that its first curvature function is ofunit length

Let 119862(119901) be a nongeodesic null curve of a Minkowskispacetime (119872 = R3

1 119892) with a Frenet frame 120585119873119882 where

119901 is a pseudo-arc-parameter and

119892 (120585119873) = 1 = 119892 (119882119882) 119892 (119873119873) = 119892 (120585 120585) = 0

(4)

To deal with the problem of nonuniqueness Cartan con-structed the following called Cartan Frenet frame for 119862(119901)

by using the above two conditions

nabla120585120585 = 119882 nabla

120585119873 = 120591119882 nabla

120585119882 = minus120591120585 minus 119873 (5)

where the torsion function 120591 is invariant up to a sign underLorentzian transformations The above frame is now calledthe null Cartan frame and the corresponding curve is the nullCartan curve (also see Bonnor [14] for the 4-dimensionalcase using Cartan method) In 1994 Bejancu [9] proved thefollowing fundamental existence and uniqueness theoremfor null curves of an (119898 + 2)-dimensional Minkowski space(119877

119898+2

1 119892) First we define in 119877119898+2

1a quasiorthonormal basis

119882119900= (minus

1

radic2

1

radic2 0 0)

1= (

1

radic2

1

radic2 0 0)

2= (0 0 1 0 0)

119898+1

= (0 0 0 1)

(6)

where 119900

1 are null vectors such that 119892(

119900

1) = 1 and

2

119898+1 are orthonormal spacelike vectors It is easy

to see that

119894

119900

119895

1+

119895

119900

119894

1+

119898+1

sum120572=2

119894

120572

119895

120572= ℎ

119894119895 (7)

Geometry 3

for any 119894 119895 isin 0 119898 + 1 where we put

ℎ119894119895

=

minus1 119894 = 119895 = 0

1 119894 = 119895 = 0

0 119894 = 119895

(8)

Theorem 1 (see [9]) Let 1198961 119896

2119898 [minus120598 120598] rarr 119877 be

everywhere continuous functions let 119909119900= (119909119894

119900) be a fixed point

of 119877119898+2

1 let and

119900

119898+1 be the quasiorthonormal basis

in (6)Then there exists a unique null curve119862 of119877119898+2

1given by

the equations119909119894

= 119909119894

(119901)119901 isin [minus120598 120598] where119901 is a distinguishedparameter on 119862 such that 119909

119894

119900= 119909

119894

(0) and 1198961 119896

2119898are

the curvature functions of 119862 with respect to a Frenet frame119865 = 119889119889119905119873119882

2 119882

119898+1 that satisfies

119889

119889119901(0) =

0 119873 (0) =

1

119882120572(0) =

120572 120572 isin 2 119898 + 1

(9)

Proof We denote nabla120585119883 by 1198831015840 Using the general Frenet

equations [6 page 55] consider the system of differentialequations

1198821015840

119900(119901) = 119896

1119882

2

1198821015840

1(119901) = 119896

2119882

2+ 119896

3119882

3

1198821015840

2(119901) = minus 119896

2119882

119900minus 119896

1119882

1+ 119896

4119882

3+ 119896

5119882

4

1198821015840

3(119901) = minus 119896

3119882

119900minus 119896

4119882

2+ 119896

6119882

4+ 119896

7119882

5

1198821015840

119898minus1(119901) = minus 119896

2119898minus5119882

119898minus3minus 119896

2119898minus4119882

119898minus2

+ 1198962119898minus3

119882119898

+ 1198962119898minus1

119882119898+1

1198821015840

119898(119901) = minus 119896

2119898minus4119882


2119898minus2119882

119898minus1+ 119896

2119898119882

119898+1

1198821015840

119898+1(119901) = 119896

2119898minus1119882


2119898119882

119898

(10)

Then there exists a unique solution 119882119900 119882

119898+1 satisfying

the initial conditions 119882119894(0) =

119894 119894 isin 0 119898 + 1

Furthermore 119882119900(119901) 119882

119898+1(119901) is a quasiorthonormal

basis such that 119882119900(119901)119882

1(119901) and 119882

2(119901) 119882

119898+1(119901) are

lightlike and spacelike respectively for each 119901 isin [minus120598 120598]Following Bonnor [14] it is proved that

119865 = 119889

119889119901= 119882

119900(119901) 119873 (119901) = 119882

1(119901)

1198822(119901) 119882

119898+1(119901)

(11)

is a Frenet frame for 119862 with curvature functions 1198961 119896

2119898

and 119901 is the distinguished parameter on 119862 which completesthe proof

In year 2001 the above theorem was generalized byFerrandez et al [15] by constructing a canonical representa-tion for nongeodesic null Cartan curves in a general (119898 + 2)-dimensional Lorentzian manifold as follows

Theorem 2 (see [15]) Let 119862(119901) be a null curve of an ori-entable Lorentzian manifold (119872119898+2

1 119892) with a pseudo-arc-

parameter 119901 such that a basis of 119879119862(119901)

119872 for all 119901 is givenby 1198621015840(119901) 11986210158401015840(119901) 119862(119898+2)(119901) Then there exists exactly oneFrenet frame 119865 = 120585119873119882

1 119882

119898 satisfying

nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus120581

1120585 minus 119873

nabla120585119882

2= minus120581

2120585 + 120581

3119882

3

nabla120585119882

119894= minus120581

119894119882

119894minus1+ 120581

119894+1119882

119894+1 119894 isin 3 119898 minus 1

nabla120585119882

119898= minus120581

119898119882

119898minus1

(12)

and fulfilling the following two conditions

(i) 1198621015840(119901) 11986210158401015840(119901) 119862(119894+2)(119901) and 1205851198731198821 119882

119894

have the same orientation for 2 le 119894 le 119898 minus 1

(ii) 1205851198731198821 119882

119898 is positively oriented and 120581

119894gt 0 for

all 119894 ge 2

The above Frenet frame of the equations its curvaturefunctions and the corresponding curve 119862 are called theCartan frame the Cartan curvatures and the null Cartancurve respectively

Corollary 3 The Cartan curvatures of a curve 119862 in119872119898+2

1are

invariant under Lorentzian transformations

Example 4 Let 119862(119901) = (1radic3)(radic2 sinh119901 radic2 cosh119901 sin 119901cos119901 119901) be a null curve of 1198775

1with vector fields of its Frenet

frame 119865 given by

120585 =1

radic3(radic2 cosh119901radic2 sinh119901 cos119901 minus sin119901 1)

119873 = minus1

6radic3(5radic2 cosh119901 5radic2 sinh119901 minus7 cos119901 7 sin119901 minus1)

1198821=

1

radic3(radic2 sinh119901radic2 cosh119901 minus sin119901 minus cos119901 0)

1198822=

1

radic6(radic2 sinh119901radic2 cosh119901 2 sin119901 2 cos119901 0)

1198823= minus

1

3radic2(radic2 cosh119901radic2 sinh119901 minus2 cos119901 2 sin119901 4)

(13)

4 Geometry

Then it is easy to obtain the following Frenet equation

nabla120585120585 = 119882

1 nabla

120585119873 = minus

1

6119882

1minus

4

3radic2119882

2

nabla120585119882

1=

1

6120585 minus 119873 nabla

120585119882

2=

4

3radic2120585 +

radic2

3119882

3

nabla120585119882

3= minus

radic2

3119882

2

(14)

Remark 5 Each null Cartan curve is a canonical repre-sentation for nongeodesic null curves For a collection ofpapers on the use of null Cartan curves soliton solutions[16] null Cartan helices and relativistic particles involvingthe curvature of 3 and 4 dimensional null curves and theirgeometricphysical applications we refer to website of Lucas[17] and a Duggal and Jin book [7]

22 Null Bertrand Curves A curve 119862 in R3 parameterizedby the arclength is a Bertrand curve [18 page 41] if and onlyif 119862 is a plane curve or its curvature and torsion are in alinear relation with constant coefficients In 2003 Honda andInoguchi [19] studied a pair of null curves (119862(119901) 119862(119901)) calleda null Bertrand pair which is defined as follows Let 120585119873119882

1

and 1205851198731198821 be the Frenet frame of 119862(119901) and 119862(119901) where

119901 and 119901 are their respective pseudo-arc-parametersThis pair(119862 119862) is said to be a null Bertrand pair (with 119862 being aBertrand mate of 119862 and vice versa) if 119882

1and 119882

1are linearly

dependentThey established a relation of Bertrand pairs withnull helices in R3

1 Recently Inoguchi and Lee have published

the following result on the existence of a Bertrand mate

Theorem 6 (see [20]) Let 119862(119901) be a null Cartan curve in R3

1

where 119901 is a pseudo-arc-parameter Then 119862 admits a Bertrandmate 119862 if and only if 119862 and 119862 have same nonzero constantcurvatures Moreover 119862 is congruent to 119862

Example 7 Let 119862(119901) = (119901 cos119901 sin119901) be a null helix withFrenet frame

119865 = 1198621015840

119873 =1

2(minus1 minus sin119901 minus cos119901)

119882 = (0 minus cos119901 minus sin119901)

(15)

It is easy to see that its torsion 120591 = 12 Define 119862 = 119862 + 2119882

with 119901 = 119901 Then 119862 = (119901 minus cos119901 minus sin119901) Therefore 119862 iscongruent to the original curve 119862

In 2005 Coken and Ciftci [21] generalized the R3

1case

of Bertrand curves for the Minkowski space R4

1using the

following Frenet equations (see (24))

nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus 120581

1120585 minus 119873

nabla120585119882

2= minus 120581

2120585

(16)

for a Frenet frame 1205851198731198821119882

2 Following is their charac-

terization theorem

Theorem 8 (see [21]) A null Cartan curve inR4

1is a Bertrand

null curve if and only if 1205811is nonzero constant and 120581

2is zero

In a recent paper Mehmet and Sadik [22] have shownthat a null Cartan curve in R4

1is not a Bertrand curve if the

derivative vectors 1198621015840 11986210158401015840 119862(3) 119862(4) (see Theorem 2) of thecurve are linearly independent

Remark 9 In Theorem 2 Ferrandez et al [15] assumed thelinear independence of the derivative vectors of the curve toobtain a unique Cartan frame However in 2010 Sakaki [23]proved that the assumption in thisTheorem 2 can be lessenedfor obtaining a unique Cartan frame in R119899

1

Open Problem We have seen in this section that the studyon Bertrand curves is focused on 3- and 4-dimensionalMinkowski spaces Theorem 2 of Ferrandez et al [15] onunique existence of null Cartan curves in a Lorentzianmanifold has opened the possibility of research on nullBertrand curves and null Bertrand mates in a 3- 4- and also119899-dimensional Lorentzian manifold and their relation withthe corresponding unique null Cartan curves

23 Null Curves in Semi-Riemannian Manifolds of Index119902 gt 1 Let 119862 be a null curve of a semi-Riemannian manifold(119872

119898+2

119902 119892) of index 119902 gt 1 Its screen distribution 119878(119879119862perp)

is semi-Riemannian of index 119902 minus 1 Therefore contrary tothe case of 119902 = 1 any of its base vector 119882

1 119882

119898

might change its causal character on U sub 119862 This opens thepossibility of more than one type of Frenet equations To dealwith this possibility in 1999 Duggal and Jin [24] studied thefollowing two types of Frenet frames for 119902 = 2

Type 1 Frenet Frames for 119902 = 2 For a null curve119862 of119872119898+2

2 any

of its screen distributions is Lorentzian Denote its generalFrenet frame by

1198651= 120585119873119882

1 119882

119898 (17)

when one of 119882119894is timelike Call 119865

1a Frenet frame of Type 1

Similar to the case of 119902 = 1 we have the following generalFrenet equations of Type 1

nabla120585120585 = ℎ120585 + 120581

1119882

1

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2

1205981nabla120585119882

1= minus 120581

2120585 minus 120581

1119873 + 120581

4119882

2+ 120581

5119882

3

Geometry 5

1205982nabla120585119882

2= minus 120581

3120585 minus 120581

4119882

1+ 120581

6119882

3+ 120581

7119882

4

1205983nabla120585119882

3= minus 120581

5119882

1minus 120581

6119882

2+ 120581

8119882

4+ 120581

9119882

5

120598119898minus1

nabla120585119882

119898minus1= minus 120581

2119898minus3119882

119898minus3

minus 1205812119898minus2

119882119898minus2

+ 1205812119898

119882119898

120598119898nabla120585119882

119898= minus 120581

2119898minus1119882


2119898119882

119898minus1

(18)

where ℎ and 1205811 120581

2119898 are smooth functions on U

1198821 119882

119898 is an orthonormal basis of Γ(119878(119879119862

perp

)|U) and

(1205981 120598

119898) is the signature of the manifold 119872

119898+2

2such that

120598119894120575119894119895

= 119892(119882119894119882

119895) The functions 120581

1 120581

2119898 are called

curvature functions of 119862 with respect to 1198651

Example 10 Let 119862 be a null curve in R6

2given by

119862 (cosh 1199051

271199053

+ 21199051

31199052

1

271199053

radic3119905 sinh 119905) 119905 isin R

(19)

Choose the following general Type 1 frame 119865 =

1205851198731198821119882

2119882

3119882

4

120585 = (sinh 1199051199052

9+ 2

2

3119905

1199052

9radic3 cosh 119905)

119873 =1

2(sinh 119905 minus

1199052

9minus 2 minus

2

3119905 minus

1199052

9 minusradic3 cosh 119905)

1198821= minus

3

radic5(cosh 119905

2

9119905

2

32

9119905 0 sinh 119905)

1198822= minus

1

radic5(2 cosh 119905 119905 3 119905 0 2 sinh 119905)

1198823= minus(0

1199052

9+

3

22

3119905

1199052

9minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(20)

Using the general Frenet equations we obtain

nabla120585120585 = minus

radic5

3119882

1 nabla

120585119873 = minus

13

6radic5119882

1+

2

radic5119882

2

1205981nabla120585119882

1=

13

6radic5120585 +

radic5

3119873 +

4

3radic5119882

3

1205982nabla120585119882

2= minus

2

radic5120585 minus

2

radic5119882

3

1205983nabla120585119882

3= minus

4

3radic5119882

1+

2

radic5119882

2

1205983nabla120585119882

4= 0 ℎ = 0

1205811= minus

radic5

3 120581

2= minus

13

6radic5

1205813=

2

radic5 120581

4= 0 120581

5=

4

3radic5

1205816= minus

2

radic5 120581

7= 0 120581

8= 0

(21)

Type 2 Frenet Frames for 119902 = 2 Construct a quasiorthonormalbasis consisting of the two null vector fields 120585 and 119873 andanother two null vector fields 119871

119894and 119871

119894+1such that

119871119894=

119882119894+ 119882

119894+1

radic2 119871

119894+1=

119882119894+1

minus 119882119894

radic2

119892 (119871119894 119871

119894+1) = 1

(22)

where119882119894and119882

119894+1are timelike and spacelike respectively all

taken from 1198651 The remaining (119898 minus 2) subset 119882

120572 of 119865

1has

all spacelike vector fields There are (119898 minus 1) choices for 119871119894for

a Frenet frame of the form

1198652= 120585119873119882

1 119871

119894 119871

119894+1 119882

119898 (23)

Denote 1198652Frame by Type 2 Following exactly as in the

previous case we have the following general Frenet equationsof Type 2

nabla120585120585 = ℎ120585 + 120581

1119882

1+ 120591

1119882

2

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2+ 120591

2119882

3

nabla120585119882

1= 120581

2120585 + 120581

1119873 minus 120581

4119882

2minus 120581

5119882

3minus 120591

3119882

4

nabla120585119882

2= minus120581

3120585 minus 120591

1119873 minus 120581

4119882

1

+ 1205816119882

3+ 120581

7119882

4+ 120591

4119882

5

nabla120585119882

3= minus120591

2120585 minus 120581

5119882

1minus 120581

6119882

2

+ 1205818119882

4+ 120581

9119882

5+ 120591

5119882

6

nabla120585119882

4= minus120591

3119882

1minus 120581

7119882

2minus 120581

8119882

3

+ 12058110119882

5+ 120581

11119882

6+ 120591

6119882

7

nabla120585119882

119898minus1= minus120591

119898minus2119882


2119898minus3119882

119898minus3


119882119898minus2

+ 1198962119898

119882119898

nabla120585119882

119898= minus120591

119898minus1119882


2119898minus1119882

119898minus2

minus 1198962119898

119882119898minus1

(24)

6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)

Choose a Frenet frame 1198652= 120585119873 119871

1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052

4minus 2 minus119905 minus

1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1

radic2(minus cosh 119905

119905

2 1

119905

2 0 minus sinh 119905)

1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)

It is easy to obtain the following Frenet equations for theabove frame 119865

2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712

nabla1205851198711= minus

1

radic2120585 +

1

radic21198711

nabla1205851198712= minusradic2119873 +

1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)

Remark 12 Note that there are119898 and119898minus 1 different choicesof constructing Frenet frames and their Frenet equations ofType 1 and Type 2 respectivelyMoreover Type 2 is preferableas it is invariant with respect to the change of its causalcharacter onU sub 119862 Also see [25] on null curves of 119877119898+2

2

Frenet Frames of Type 119902(ge 3) Using the above procedure wefirst construct Frenet frames of null curves 119862 in 119872119898+2

3 Their

screen distribution 119878(119879119862perp) is of index 2 Therefore we have2 timelike vector fields in 119882

1 119882

119898 To understand this

take a case when 1198821119882

2 are timelike The construction of

Type 1 and Type 2 frames is exactly the same as that in thecase 119902 = 2 so we give details for Type 3 Transform the Frenetframe 119865

1of Type 1 into another frame which consists of two

null vector fields 120585 and119873 and additional four null vector fields1198711 119871

2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)

The remaining (119898 minus 4) vector fields of subset 119882120572 of 119865

1are

all spacelike In this case we have a Frenet frame of the form

1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)

Denote1198653frame byType 3which is preferable choice as any of

its vector fields will not change its causal character onU sub 119862In this way one can use all possible choices of two timelikevector fields from 119865

1and construct corresponding forms of

Frenet frames of Type 3

The above procedure can be easily generalized to showthat the null curves of 119872119898+2

119902have 119865

119902Frenet frames of Type

1 Type 2 Type 119902 Also there are a variety of each of suchtype and their corresponding Frenet equations

Precisely if 119902 = 1 then119872119898+2

1has Type 1 of Frenet frames

if 119902 = 2 then 119872119898+2

2has two types of Frenet frames labeled

Type 1 and Type 2 up to the signs of 119882119894 and if 119902 = 119899 then

119872119898+2

119899have 119899-types labeled Type 1 Type 2 Type 119899 up

to the signs of 119882119894 However for each 119902 = 119899 gt 1 only one

frame of Type 119899 will be a preferable frame as it is invariantwith respect to the change of its causal character onU sub 119862


3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)

with Type 3 Frenet frame 1198653= 120585119873 119871

1 119871

2 119871

3 119871

4 given as

follows120585 = (minus sin 119905 cos 119905 cosh 119905 sinh 119905 1 1)

119873 =1

2(sin 119905 minus cos 119905 minus cosh 119905 minus sinh 119905 1 minus1)

1198711= (minus cos 119905 minus sin 119905 sinh 119905 cosh 119905 0 0)

1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)

Then one can calculate its following Frenet equations

nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7

Related to the focus of this paper we now discuss the issueof unique existence of null curves in a semi-Euclidean space(119877

119898+2

119902 119892)

Fundamental Theorems of Unique Null Curves in 119877119898+2

119902 Sup-

pose 119862 is a null curve in (119877119898+2

119902 119892) locally given by

119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)

with a semi-Euclidean metric

119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882

119898 be its general Frenet frame

of Type 1 Take 1198821 119882

119902minus1 timelike and the rest (119898 minus

119902 minus 1) vector fields 119882119902 119882

119898 spacelike There is one

fundamental theorem for each of 119902-types We give details forthe last Type 119902 whose Frenet frame is given by

119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)

where its vector fields are defined by

1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast

119902 are null vector fields such that

119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)

In this case we find119902minus1

sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)

for any 119860 119861 isin 0 (119898 + 1) where we put

ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)

Now we state the following fundamental existence anduniqueness theorem for null curves of 119877119898+2

119902(which also

includes Theorem 1 for the case 119902 = 1)

Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877

be everywhere continuous functions 119909119900

= (119909119894

119900) a fixed point

of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898

the quasiorthonormal basis of a Frenet frame 119865119902as displayed

above Then there exists a unique null curve 119862 [minus120576 120576] rarr

119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894

are curvature and torsion functions with respect to this Frenetframe 119865

119902of Type 119902 satisfying

120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)

The construction of the Frenet equations is similar to Frenetequations (10) of Theorem 1 and the rest of the proof easilyfollows

We refer to [7 Chapters 2ndash4] for proofs of the fundamen-tal theorems the geometry of all possible types of null curvesin 119872

119898+2

119902 and many examples

Open Problem In previous presentation we have seen thatcontrary to the nondegenerate case the uniqueness of anytype of general Frenet equations cannot be assured even ifone chooses a pseudo-arc-parameter Each type depends onthe parameter of 119862 and the choice of a screen distributionHowever for a null curve in a Lorentzian manifold using thenatural Frenet equations we found a unique Cartan Frenetframe whose Frenet equations have a minimum number ofcurvature functions which are invariant under Lorentziantransformations This raises the following question is thereexist any unique Frenet frame for null curves in a general semi-Riemannian manifold (M119898+2

119902 119892) We therefore invite the

readers to work on the following research problemFind condition(s) for the existence of unique Frenet

frames of nongeodesic null curves in a semi-Riemannianmanifold of index 119902 where 119902 gt 2

3 Unique or Canonical Theorems inLightlike Hypersurfaces

Let (119872 119892) be a hypersurface of a prope (119898 + 2)-dimensi-onal semi-Riemannian manifold (119872 119892) of constant index

8 Geometry

119902 isin 1 119898+1 Suppose 119892 is degenerate on119872 Then thereexists a vector field 120585 = 0 on 119872 such that 119892(120585 119883) = 0 for all119883 isin Γ(119879119872) The radical subspace Rad119879

119909119872 of 119879

119909119872 at each

point 119909 isin 119872 is defined by

Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)

dim(Rad119879119909119872) = 1 and (119872 119892) is called a lightlike hypersur-

face of (119872 119892) We call Rad119879119872 a radical distribution of 119872Since 119879119872 sup 119879119872perp contrary to the nondegenerate case theirsum is not the whole of tangent bundle space 119879119872 In otherwords a vector of 119879

119909119872 cannot be decomposed uniquely into

a component tangent to 119879119909119872 and a component of 119879

119909119872perp

Therefore the standard text-book definition of the secondfundamental form and the Gauss-Weingarten formulas donot work for the lightlike case To deal with this problemin 1991 Bejancu and Duggal [10] introduced a geometrictechnique by splitting the tangent bundle 119879119872 into twononintersecting complementary (but not orthogonal) vectorbundles (one null and one nonnull) as follows Consider acomplementary vector bundle 119878(119879119872) of 119879119872

perp = Rad119879119872 in119879119872 This means that

119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)

where 119878(119879119872) is called a screen distribution on 119872 whichis nondegenerate Thus along 119872 we have the followingdecomposition

119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)

that is 119878(119879119872)perp is orthogonal complement to 119878(119879119872) in119879119872

|119872

which is also nondegenerate but it includes Rad119879119872 as its subbundle We need the following taken from [6 Chapter 4]

There exists a unique vector bundle tr(119879119872) of rank 1

over 119872 such that for any nonzero section 120585 of Rad119879119872 ona coordinate neighborhoodU sub 119872 we have a unique section119873 of tr(119879119872) onU satisfying

119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)

It follows that tr(119879119872) is lightlike such that tr(119879119872)119906cap119879

119906119872 =

0 for any 119906 isin 119872 Moreover we have the followingdecompositions

119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)

Hence for any screen distribution 119878(119879119872) there is a uniquetr(119879119872) which is complementary vector bundle to 119879119872 in119879119872

|119872 called the lightlike transversal vector bundle of 119872

with respect to 119878(119879119872) Denote by (119872119892 119878(119879119872)) a lightlikehypersurface of (119872 119892) The Gauss and Weingarten typeequations are

nabla119883119884 = nabla

119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)

respectively where 119861 is the local second fundamental formof 119872 and 119860

119873is its shape operator It is easy to see that

119861(119883 120585) = 0 for all 119883 isin Γ(119879119872|U) Therefore 119861 is degenerate

with respect to 119892 Moreover the connection nabla on 119872 is not ametric connection and satisfies

(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)

In the lightlike case we also have another second fundamen-tal form and its corresponding shape operator which we nowexplain as follows

Let 119875 denote the projection morphism of Γ(119879119872) onΓ(119878(119879119872)) We obtain

nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast

120585119883 minus 120591 (119883) 120585 forall119883 119884 isin Γ (119879119872)

(48)

where 119862(119883 119875119884) is the screen fundamental form of 119878(119879119872)The two second fundamental forms of 119872 and 119878(119879119872) arerelated to their shape operators by

119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)

31 Unique or Canonical Screen Distributions Unfortunatelythe induced objects (second fundamental forms inducedconnection structure equations etc) depend on the choice ofa screen 119878(119879119872) which in general is not unique This raisesthe question of finding those lightlike hypersurfaces whichadmit a unique or canonical 119878(119879119872) needed in the lightlikegeometry Although a positive answer to this question for anarbitrary 119872 is not possible due to the degenerate inducedmetric considerable progress has been made to heal thisessential anomaly for specific classes

There are several approaches in dealing with thisnonuniqueness problem Some authors have used specificmethods suitable for their problems For example Akivis andGoldberg [26ndash28] Bonnor [29] Leistner [30] Bolos [31] aresamples of many more authors in the literature Also Kupeli[5] has shown that 119878(119879119872) is canonically isometric to thefactor vector bundle 119879119872

lowast = 119879119872119879119872perp and used canonicalprojection 120587 119879119872 rarr 119879119872lowast in studying the intrinsic geo-metry of degenerate semi-Riemannian manifolds where ourreview in this paper is focused on the extrinsic geometrywhich is in line with the classical theory of submanifolds[1] Consequently although specific techniques are suitablefor good applicable results nevertheless for the fundamentaldeeper study of extrinsic geometry of lightlike spaces onemust look for a canonical or a unique screen distributionFor this purpose we first start with a chronological historyof some isolated results and then quote two main theorems

In 1993 Bejancu [32] constructed a canonical 119878(119879119872)

for lightlike hypersurfaces 119872 of semi-Euclidean spaces 119877119899+1

119902

Then in [10 Chapter 4] it was proved that such a canonicalscreen distribution is integrable on any lightlike hypersurfaceof119877119899

1and on any lightlike cone and119899

119902minus1of119877119899+1

119902This information

was used by Bejancu et al [33] in showing some interested

Geometry 9

geometric results Later on Akivis andGoldberg [28] pointedout that such a canonical construction was neither invari-ant nor intrinsically connected with the geometry of 119872Therefore in the same paper [28] they constructed invariantnormalizations intrinsically connected with the geometryof 119872 and investigated induced linear connections by thesenormalizations using relative and absolute invariant definedby the first and second fundamental forms of 119872

Let 119865 = 120585119873119882119886 119886 isin 1 119898 be a quasiorthonormal

basis of 119872 along 119872 where 120585 119873 and 119882119886 are null

basis of Γ(Rad119879119872|U) Γ(tr(119879119872)

|U) and orthonormal basisof Γ(119878(119879119872)

|U) respectively For the same 120585 consider otherquasiorthonormal frames fields 1198651015840 = 12058511987310158401198821015840

119886 induced on

U sub 119872 by 1198781015840(119879119872) (tr)1015840(119879119872) It is easy to obtain

1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)

where 120598119886 are signatures of orthonormal basis 119882

119886 and119882119887

119886

f and f119886are smooth functions on U such that [119882119887

119886] is 119898 times

119898 semiorthogonal matrices Computing 119892(1198731015840 1198731015840) = 0 and119892(119882

119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2

= 0 Using this in thesecond relation of the above two equations we get

1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)

The above two relations are used to investigate the transfor-mation of the induced objects when the pair 119878(119879119872) tr(119879119872)

changes with respect to a change in the basis To look for acondition so that a chosen screen is invariant with respectto a change in the basis in 2004 Atindogbe and Duggalobserved that a nondegenerate hypersurface has only onefundamental form where as a lightlike hypersurface admitsan additional fundamental form of its screen distributionand their two respective shape operators Moreover we know[1] that the fundamental form and its shape operator of anondegenerate hypersurface are related by the metric tensorContrary to this we see from the two equations of (49) thatin the lightlike case there are interrelations between its twosecond fundamental forms Because of the above differencesAtindogbe and Duggal were motivated to connect the twoshape operators by a conformal factor as follows

Definition 15 (see [34]) A lightlike hypersurface (119872 119892

119878(119879119872)) of a semi-Riemannian manifold is called screenlocally conformal if the shape operators119860

119873and119860lowast

120585of119872 and

119878(119879119872) respectively are related by

119860119873

= 120593119860lowast

120585 (52)

where 120593 is a nonvanishing smooth function on a neighbor-hoodU in 119872

To avoid trivial ambiguities we take U connected andmaximal in the sense that there is no larger domainU1015840 sup Uon which the above relation holds It is easy to show that twosecond fundamental forms 119861 and 119862 of a screen conformallightlike hypersurface 119872 and its 119878(119879119872) respectively arerelated by

119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)

Denote by S1 the first derivative of 119878(119879119872) given by

S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)

Let 119878(119879119872) and 119878(119879119872)1015840 be two screen distributions on 119872

119861 and 1198611015840 their second fundamental forms with respectto tr(119879119872) and tr (119879119872)

1015840 respectively for the same 120585 isin

Γ(119879119872perp|U) Denote by 120596 the dual 1-form of the vector field119882 = sum

119898

119886=1f119886119882

119886with respect to 119892 Following is a unique

existence theorem

Theorem 16 (see [8] page 61) Let (119872 119892 119878(119879119872)) be ascreen conformal lightlike hypersurface of a semi-Riemannianmanifold (119872 119892) withS1 the first derivative of 119878(119879119872) given by(54) Then

(1) a choice of the screen 119878(119879119872) of 119872 satisfying (52) isintegrable

(2) the one form 120596 vanishes identically on S1(3) ifS1 coincides with 119878(119879119872) then119872 can admit a unique

screen distribution up to an orthogonal transformationand a unique lightlike transversal vector bundle More-over for this class of hypersurfaces the screen secondfundamental form 119862 is independent of its choice

Proof It follows from the screen conformal condition (52)that the shape operator 119860

lowast

120585of 119878(119879119872) is symmetric with

respect to 119892 Therefore a result [6 page 89] says that achoice of screen distribution of a screen conformal lightlikehypersurface 119872 is integrable which proves (1)

As 119878(119879119872) is integrableS1 is its subbundle AssumeS1 =

119878(119879119872) Then it is easy to show that 120596 vanishes on 119878(119879119872)which implies that the functions f

119886of the transformation

equations vanish Thus the transformation equation (51)becomes 1198821015840

119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887

119886) is an orthogonal matrix of 119878(119879

119909119872) at any point 119909 of119872

which proves the first part of (3) Then independence of 119862

follows which completes the proof

Remark 17 Based on the above theorem onemay ask the fol-lowing converse question Does the existence of a canonicalor a unique distribution 119878(119879119872) of a lightlike hypersurfaceimply that 119878(119879119872) is integrable Unfortunately the answerin general is negative which we support by recalling thefollowing known results from [6 pages 114ndash117]

10 Geometry

There exists a canonical screen distribution for anylightlike hypersurface of a semi-Euclidean space 119877

119898+2

119902 how-

ever only the canonical screen distribution on any lightlikehypersurface of 119877119898+2

1is integrable Therefore although any

screen conformal lightlike hypersurface admits an integrablescreen distribution the above results say that not every suchintegrable screen coincides with the corresponding canonicalscreen that is there are cases for which S1 = 119878(119879119872)

Now one may ask whether there is a class of semi-Riemannian manifolds which admit screen conformal light-like hypersurfaces and therefore can admit a unique screendistribution This question has been answered as follows

Theorem 18 (see [11]) Let (119872119892 119878(119879119872)) be a lightlike hyper-surface of a semi-Riemannian manifold (119872

119898+2

119902 119892) with 119864 a

complementary vector bundle of 119879119872perp in 119878(119879119872)perp such that 119864

admits a covariant constant timelike vector field Then withrespect to a section 120585 of Rad119879119872 119872 is screen conformal Thus119872 can admit a unique screen distribution

To get a better idea of the proof of this theorem we givethe following example

Example 19 (see [8] page 62) Consider a smooth function119865 Ω rarr 119877 where Ω is an open set of 119877119898+1 Then

119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)

is a Monge hypersurfaceThe natural parameterization on119872

is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)

Hence the natural frames field on 119872 is globally defined by

120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)

spans 119879119872perp Therefore 119872 is lightlike (ie 119879119872perp = Rad119879119872)if and only if the global vector field 120585 is spanned by Rad119879119872

which means if and only if 119865 is a solution of the partialdifferential equation

1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)

Along 119872 consider the constant timelike section 119881 = 1205971199090 of

Γ(119879119877119898+2

119902) Then 119892(119881 120585) = minus1 implies that 119881 is not tangent

to 119872 Therefore the vector bundle 119864 = Span119881 120585 is nonde-generate on 119872 The complementary orthogonal vector bun-dle 119878(119879119872) to 119864 in 119879119877119898+2

119902is a nondegenerate distribution on

119872 and is complementary to Rad119879119872 Thus 119878(119879119872) is a screen

distribution on119872The transversal bundle 119905119903(119879119872) is spannedby119873 = minus119881+(12)120585 and 120591(119883) = 0 for any119883 isin Γ(119879119872) Indeed120591(119883) = 119892(nabla

119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten

equations reduce to nabla119883119873 = minus119860

119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)

Hence any lightlike Monge hypersurface of 119877119898+2

119902is screen

globally conformal with 120593(119909) = 12 Therefore it can admit aunique screen distribution

32 Unique Metric Connection and Symmetric Ricci TensorWe know from (47) that the induced connectionnabla on a light-like submanifold (119872 119892) is a metric (Levi-Civita) connectionif and only if the second fundamental form 119861 vanishes on119872The issue is to find conditions on the induced objects of alightlike hypersurface which admit such a unique Levi-Civitaconnection First we recall the following definitions

In case any geodesic of 119872 with respect to an inducedconnection nabla is a geodesic of 119872 with respect to nabla we saythat119872 is a totally geodesic lightlike hypersurface of119872 Alsonote that a vector field 119883 on a lightlike manifold (119872 119892) issaid to be a Killing vector field if pound

119883119892 = 0 A distribution 119863

on119872 is called a Killing distribution if each vector field of119863 isKilling

Nowwe quote the following theorem on the existence of aunique metric connection on119872 which also shows from theGauss equation that the definition of totally geodesic119872 doesnot depend on the choice of a screen

Theorem 20 (see [6]) Let (119872 119892 119878(119879119872)) be a lightlikehypersurface of a semi-Riemannian manifold (119872 119892) Then thefollowing assertions are equivalent

(a) 119872 is totally geodesic in 119872(b) 119861 vanishes identically on 119872(c) 119860lowast

120585vanish on Γ(119879119872) for any 120585 isin Γ(Rad119879119872)

(d) There exists a unique torsion-free metric connection nabla

on 119872(e) Rad TM is a Killing distribution(f) Rad TM is a parallel distribution with respect to nabla

On the issue of obtaining an induced symmetric Riccitensor of 119872 we proceed as follows Consider a type (0 2)

induced tensor on 119872 given by

119877 (119883 119884) = trace 119885 997888rarr 119877 (119883119885) 119884 forall119883 119884 isin Γ (119879119872)

(61)

Let 120585119882119886 be an induced quasiorthonormal frame on 119872

where Rad 119879119872 = Span120585 and 119878(119879119872) = Span119882119886 and let

119864 = 120585119873119882119886 be the corresponding frames field on119872Then

we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11

where 120598119886denotes the causal character (plusmn1) of respective

vector field 119882119886 Using Gauss-Codazzi equations we obtain

119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)

Substituting this into the previous equation and using therelations (49) we obtain

119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)

where Ric is the Ricci tensor of 119872 This shows that 119877(119883 119884)

is not symmetric Therefore in general it has no geometricor physical meaning similar to the symmetric Ricci tensorof 119872 Thus this 119877(119883 119884) can be called an induced Riccitensor of 119872 only if it is symmetric Thus one may ask thefollowing question are there any lightlike hypersurfaces withsymmetric Ricci tensor The answer is affirmative for whichwe quote the following result

Theorem 21 (see [34]) Let (119872 119892 119878(119879119872)) be a locally (orglobally) screen conformal lightlike hypersurface of a semi-Riemannian manifold (119872(119888) 119892) of constant sectional curva-ture 119888 Then 119872 admits an induced symmetric Ricci tensor

Proof Using the curvature identity 119877(119883 119884)119885 = 119888119892(119884 119885)119883minus

119892(119883 119885)119884 and the equation in (64) we obtain

119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)

Then using the screen conformal relation (52) mentionedabove it is easy to show that 119877(119883 119884) is symmetric andtherefore it is an induced Ricci tensor of 119872

In particular if 119872 is totally geodesic in 119872 then usingthe curvature identity and proceeding similarly to what ismentioned above one can show that

119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)

Since Ric and 119892 are symmetric we conclude that any totallygeodesic lightlike hypersurface of 119872(119888) admits an inducedsymmetric Ricci tensor

Finally we quote a general result on the induced symmet-ric Ricci tensor

Theorem 22 (see [6]) Let (119872 119892 119878(119879119872)) be a lightlike hyper-surface of a semi-Riemannianmanifold (119872 119892)Then the tensor119877(119883 119884) defined in (61) of the induced connection nabla is asymmetric Ricci tensor if and only if each 1-form 120591 induced by119878(119879119872) is closed that is 119889120591 = 0 on anyU sub 119872

Remark 23 The symmetry property of the Ricci tensor on amanifold119872 equipped with an affine connection has also beenstudied by Nomizu-Sasaki In fact we quote the followingresult (Proposition 31 Chapter 1) in their 1994 book

Proposition 24 (see [35]) Let (119872 nabla) be a smooth manifoldequipped with a torsion-free affine connection Then the Riccitensor is symmetric if and only if there exists a volume element120596 satisfying nabla120596 = 0

Open Problem Give an interpretation ofTheorem 22 in termsof affine geometry

33 Induced Scalar Curvature To introduce a concept ofinduced scalar curvature for a lightlike hypersurface 119872 weobserve that in general the nonuniqueness of screen distri-bution 119878(119879119872) and its nondegenerate causal structure rule outthe possibility of a definition for an arbitrary 119872 of a semi-Riemannian manifold Although now there are many casesof a canonical or unique screen and canonical transversalvector bundle the problem of scalar curvature must beclassified subject to the causal structure of a screen For thisreason work has been done on lightlike hypersurfaces 119872 ofa Lorentzian manifold (119872 119892) for which we know that anychoice of its screen 119878(119879119872) is Riemannian This case is alsophysically useful To calculate an induced scalar function 119903

by setting119883 = 119884 = 120585 and then119883 = 119884 = 119882119886in (62) and using

Gauss-Codazzi equations we obtain

119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)

In general 119903 given by the above expression cannot be calleda scalar curvature of 119872 since it has been calculated from atensor quantity119877(119883 119884) It can only have a geometricmeaningif it is symmetric and its value is independent of the screenits transversal vector bundle and the null section 120585 Thus torecover a scalar curvature we recall the following conditions[36] on 119872

A lightlike hypersurface 119872 (labeled by 1198720) of a

Lorentzian manifold (119872 119892) is of genus zero with screen119878(119879119872)

0 if(a) 119872 admits a canonical or unique screen distribution

119878(119879119872) that induces a canonical or unique lightliketransversal vector bundle 119873

(b) 119872 admits an induced symmetric Ricci tensordenoted by Ric

Denote by C[119872]0= [(119872 119892 119878(119879119872))]

0 a class of lightlikehypersurfaces which satisfy the above two conditions

12 Geometry

Definition 25 Let (1198720 1198920 1205850 1198730) belong to C[119872]0 Then

the scalar 119903 given by (67) is called its induced scalarcurvature of genus zero

It follows from (a) that 119878(119879119872)0 and 1198730 are either

canonical or unique For the stability of 119903 with respect to achoice of the second fundamental form 119861 and the 1-form120591 it is easy to show that with canonical or unique 119873

0 both119861 and 120591 are independent of the choice of 1205850 except for anonzero constant factor Finally we know [8 page 70] thatthe Ricci tensor does not depend on the choice of 1205850 Thus 119903is awell-defined induced scalar curvature of a class of lightlikehypersurfaces of genus zero The following result shows thatthere exists a variety of Lorentzian manifolds which admithypersurfaces of classC[119872]

0

Theorem 26 (see [36]) Let (119872 119892 119878(119879119872)) be a screenconformal lightlike hypersurface of a Lorentzian space form(119872(119888) 119892) with S1 the first derivative of 119878(119879119872) given by (54)If S1 = 119878(119879119872) then 119872 belongs to C[119872]

0 Consequently thisclass of lightlike hypersurfaces admits induced scalar curvatureof genus zero

Since S1 = 119878(119879119872) it follows from Theorem 16 that 119872

admits a unique screen distribution 119878(119879119872) that induces aunique lightlike transversal vector bundle which satisfies thecondition (a) The condition (b) also holds fromTheorem 21

Moreover consider a class of Lorentzianmanifolds (119872 119892)

which admit at least one covariant constant timelike vectorfield Then Theorem 18 says that 119872 belongs to C[119872]

0 andtherefore it admits an induced scalar curvature Also see [3738] for a followup on scalar curvature

Remark 27 We know from Duggal and Bejancursquos book [6Page 111] that any lightlike surface 119872 of a 3-dimensionalLorentz manifold is either totally umbilical or totallygeodesic Moreover there exists a canonical screen distri-bution 119878(119879119872) for a lightlike Monge surface (119872 119892 119878(119879119872))

of 1198773

1for which the induced linear connection is flat (see

Proposition 71 on page 126 in [6]) In particular the null coneof 1198773

1has flat induced connection Readers may get more

information on pages 123ndash138 on lightlike hypersurfaces of1198773

11198774

1 and1198774

2in [6] In general see Section 91 of Chapter 9 in

a book by Duggal and Sahin [8] on null surfaces of spacetimes

4 Unique Existence Theorems inLightlike Submanifolds of Index 119902 gt 1

Let (119872 119892) be a real (119898 + 119899)-dimensional proper semi-Riemannian manifold where 119898 gt 1 119899 gt 1 with 119892 a semi-Riemannianmetric on119872 of constant index 119902 isin 2 119898+119899minus

1 Hence119872 is never a RiemannianmanifoldThe conditions119898 gt 1 and 119899 gt 1 imply that 119872 is neither a curve nor ahypersurface

119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)

For a lightlike119872 there exists a smooth distribution such that

Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)

The following are four cases of lightlike submanifolds

(A) 119903-lightlike submanifold 0 lt 119903 lt min119898 119899

(B) coisotropic submanifold 1 lt 119903 = 119899 lt 119898

(C) isotropic submanifold 1 lt 119903 = 119898 lt 119899

(D) totally lightlike submanifold 1 lt 119903 = 119898 = 119899

We follow [8] for notations For the case (A) of 119903-lightlikesubmanifold there exist two complementary nondegeneratedistributions 119878(119879119872) and 119878(119879119872perp) of Rad(119879119872) in 119879119872 and119879119872perp respectively called the screen and screen transversaldistributions on 119872 such that

119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp

= Rad (119879119872)oplusorth119878 (119879119872perp

) (70)

Let tr(119879119872) and ltr(119879119872) be complementary (but not orthog-onal) vector bundles to 119879119872 in 119879119872

|119872and 119879119872

perp in 119878(119879119872)perp

respectively and let 119873119894 be a lightlike basis of Γ(ltr(119879119872)

|U)

consisting of smooth sections of 119878(119879119872)perp

|U where U is a

coordinate neighborhood of 119872 such that

119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585

119903 is a lightlike basis of Γ(Rad(119879119872)) Then

119879119872 = 119879119872 oplus tr (119879119872)

= Rad (119879119872) oplus tr (119879119872) oplusorth119878 (119879119872)

= Rad (119879119872) oplus ltr (119879119872) oplusorth119878 (119879119872)oplusorth119878 (119879119872perp

)

(72)

For the case (B) of coisotropic submanifold Rad119879119872 = 119879119872perp

implies that 119878(119879119872)perp

= 0 For the cases (C) and (D)119878(119879119872) = 0 Thus to review the unique results on screen119878(119879119872) we only deal with the first two cases

119903-Lightlike Submanifolds (119872 119892 119878(119879119872) 119878(119879119872)perp

) Considerthe following local quasiorthonormal frame of 119872 along 119872

1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882

119899 are orthonormal

basis of Γ(119878(119879119872)) and Γ(119878(119879119872perp)) Let nabla be the Levi-Civitaconnection of 119872 and 119875 the projection morphism of Γ(119879119872)

Geometry 13

on Γ(119878(119879119872)) For a lightlike submanifold the local Gauss-Weingartan formulas are given by


119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)

where nabla and nablalowast are induced linear connections on 119879119872

and 119878(119879119872) respectively ℎℓ119894and ℎ119904

120572 given in (74) are called

the local transversal second fundamental forms and screensecond fundamental forms on 119879119872 respectively ℎlowast

119894 given in

(77) are called the local radical second fundamental formson 119878(119879119872)119860

119873119894

119860119882120572

and119860lowast

120585119894

are linear operators on Γ(119879119872)and 120591

119894119895 120588

119894120572 120601

120572119894 and 120579

120572120573are 1-forms on 119879119872 as given in (75)

(76) and (78) Since the connectionnabla is torsion-freenabla is alsotorsion-free connection and both ℎℓ

119894and ℎ119904

120572are symmetric

From the fact that ℎℓ119894(119883 119884) = 119892(nabla

119883119884 120585

119894) we know that ℎℓ

119894

are independent of the choice of a screen distribution Notethat ℎℓ

119894 120591

119894119895 and 120588

119894120572depend on the sections 120585 isin Γ(Rad(119879119872)|U)

and 119889(tr(120591119894119895)) = 119889(tr(120591

119894119895)) The induced connection nabla on 119879119872

is not metric connection as it satisfies

(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)

where 120578119894are the 1-forms 120578

119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)

119894 isin 1 119903However the connection nablalowast on 119878(119879119872) is metric The

above types of second fundamental forms of 119872 and 119878(119879119872)

are related as follows

ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)

Consider two quasiorthonormal frames 120585119894 119873

119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)

1198781015840(119879119872perp) 1198651015840 where 119865 and 1198651015840 are the complementary vectorbundles of Rad119879119872 in 119878(119879119872

perp

)perp and 119878

1015840

(119879119872perp

)perp respectively

Consider the following transformation equations (see [6pages 163ndash165])

1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598

120572 are signatures of bases 119883

119886 and 119882

120572

respectively 119883119887

119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876

119894120572are smooth functions

on U such that [119883119887

119886] and [119882120573

120572] are (119898 minus 119903) times (119898 minus 119903) and

(119899 minus 119903) times (119899 minus 119903) semiorthogonal matrices and

119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry

Lemma 28 (see [8]) The second fundamental forms ℎlowast andℎ1015840lowast of screens 119878(119879119872) and 119878(119879119872)

1015840 respectively in an 119903-lightlikesubmanifold 119872 are related as follows

ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572

Let 120596119894be the respective 119899 dual 1-forms of Z

119894given by

120596119894(119883) = 119892 (119883Z

119894) forall119883 isin Γ (119879119872) 1 le 119894 le 119899 (88)

Denote by S the first derivative of a screen distribution119878(119879119872) given by

S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)

If 119878(119879119872) is integrable then S is a subbundle of 119878(119879119872) At apoint 119909 isin 119872 let N1(119909) be the space spanned by all vectorsℎ119904

(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)

Call N1 as first screen transversal space We see from (80)and (83) that there are interrelations between the secondfundamental forms of the lightlike119872 and its screen distribu-tion and their respective shape operators This interrelationindicates that the lightlike geometry depends on a choiceof screen distribution which is not unique While we knowfrom (86) that the fundamental forms ℎ

ℓ

119894of the lightlike 119872

are independent of a screen the same is not true for thefundamental forms ℎ

lowast

119894of 119878(119879119872)(see (87)) which is the root

of nonuniqueness anomaly in the 119903-lightlike geometry Thequestion is how to proceed further to get the choice of aunique screen In the case of hypersurfaces we related thetwo shape operators119860


120585by a conformal function (see

(52)) However this condition cannot be used for the case ofgeneral submanifolds for the following reason

The single shape operator119860119873of a hypersurface is 119878(119879119872)-

valued but for a submanifold there is no guarantee that each119860

119873119894

is 119878(119879119872)-valued Thus shape operators cannot be usedto define screen conformal condition For this reason Duggaland Sahin [8] modified the screen conformal condition asfollows

Definition 29 A lightlike submanifold (119872 119892 119878(119879119872)

119878(119879119872perp)) of a semi-Riemannian manifold (119872 119892 is calleda screen conformal submanifold if the fundamental forms

ℎlowast

119894of 119878(119879119872) are conformally related to the corresponding

lightlike fundamental forms ℎ119894of 119872 by

ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)

where 120593119894119904 are smooth function on a neighborhoodU in 119872

In order to avoid trivial ambiguities wewill considerU tobe connected andmaximal in the sense that there is no largerdomain U1015840 sup U on which (91) holds In case U = 119872 thescreen conformality is said to be global Since this materialis relatively new for the benefit of interested readers we givecomplete proof of the following unique existence theorem

Theorem 30 (see [8]) Let (119872 119892 119878(119879119872) 119878(119879119872perp)) be an

119898-dimensional 119903-lightlike screen conformal submanifold of asemi-Riemannian manifold 119872

119898+119899 Then

(a) a choice of a screen distribution of119872 satisfying (91) isintegrable

(b) all the 119899-forms 120596119894in (89) vanish identically on the first

derivative S given by (88)(c) if S = 119878(119879119872) and N1 = 119878(119879119872perp) then there exists

a set of 119903 null sections 1205851 120585

119903 of Γ(Rad119879119872) with

respect to 119878(119879119872) which is a unique screen distributionof 119872 up to an orthogonal transformation with aunique set 119873

1 119873

119903 of lightlike transversal vector

bundles and the screen fundamental forms ℎlowast119894are

independent of a screen distribution

Proof Substituting (91) in (75) and then using (77) we get

119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894

is symmetric with respect to 119892 the aboveequation implies that each 119860

119873119894

is self-adjoint on Γ(119878(119879119872))

with respect to 119892 which further follows from [6 page 161]that a choice of a screen 119878(119879119872) satisfying (91) is integrableThus (a) holds Now choose an integrable screen 119878(119879119872) suchthat S = 119878(119879119872) Then one can obtain that all f

119894119886= 0 Using

Lemma 28 proceeding closer to the case of hypersurfaces(see Theorem 16) each W

119894and all 119876

119894120572vanish which proves

the three statements of the theorem

There exists another class of 119903-lightlike submanifoldswhich admit integrable unique screen distributions subjectto a geometric condition (different from the screen conformalcondition) which we present as follows Consider a comple-mentary vector bundle 119865 of Rad119879119872 in 119878(119879119872

perp)perp and choose

a basis 119881119894 119894 isin 1 119903 of Γ(119865

|U) Thus we are looking forthe following sections

119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861

119894119896are smooth functions on U Then 119873

119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881

119896) 119895 119896 isin 1 119903 Observe that 119866 = det[119892

119895119896]

is everywhere nonzero on U otherwise 119878(119879119872perp)perp would be

degenerate at least at a point ofU

Theorem 31 (see [8]) Let (119872 119892 119878(119879119872) 119878(119879119872perp)) be a 119903-lightlike submanifold of a semi-Riemannian manifold (119872 119892)

such that the complementary vector bundle 119865 of Rad119879119872 in119878(119879119872

perp)perp is parallel along the tangent direction Then all the

assertions from (a) through (c) of Theorem 30 will hold

Proof Covariant derivative of (93) provides

nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)

Using (74) (75) and (78) we obtain

119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896

times minus119860lowast

120585119896

119883 + nablalowast119905

119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)

Since 119865 is parallel nabla119883119881119896isin Γ(119865) Thus for 119884 isin Γ(119878(119879119872)) we

get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896

119883119884) Then we obtain

119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)

Since the right side of (96) is symmetric it follows that ℎlowast

is symmetric on 119878(119879119872) Thus it follows from [6 page 161]that 119878(119879119872) is integrableThe rest of the proof is similar to theproof of Theorem 30

Remark 32 In [8] it has been shown by an example that thescreen conformal condition does not necessarily imply that 119865is parallel along the tangent direction Thus the two classesof lightlike submanifolds with the choice of unique screenare different from each other Finally one can verify that theabove two theorems will also hold for a subcase of coisotropicsubmanifold for which 119878(119879119872

perp) vanishes and 119903 = 119899 Detailsmay be seen in [39 40]

41 Unique Metric Connection and Symmetric Ricci TensorWe know from (79) that the induced connection nabla on alightlike submanifold (119872 119892) is a unique metric (Levi-Civita)connection if and only if the second fundamental forms ℎℓ

119894

vanish on 119872 Also any totally geodesic 119872 admits a Levi-Civita connection It is important to note that contrary tothe case of hypersurfaces (seeTheorem 20) an 119903-lightlike119872with a unique metric connection is not necessarily totallygeodesic as 119878(119879119872

perp) need not vanish on 119872 However this

equivalence holds for a coisotropic 119872 for which 119878(119879119872perp)

vanishes Thus the issue is to find a variety of nontotallygeodesic lightlike submanifolds which admit such a uniqueLevi-Civita connection On this issue we quote the followingrecent result

Theorem 33 Let (119872 119892 119878(119879119872)) be an 119903-lightlike orcoisotropic submanifold of a semi-Riemannian manifold119872 such that 119878(119879119872) is Killing and 119872 is irrotational Then theinduced connection nabla on 119872 is a Levi-Civita connection

Proof Recall that a lightlike submanifold 119872 is irrotational ifnabla119883120585 isin Γ(119879119872) for any 119883 isin Γ(119879119872) where 120585 isin Γ(Rad119879119872) [5]

This further means that ℎ119897(119883 120585) = 0 and ℎ119904(119883 120585) = 0 for all120585 isin Γ(Rad119879119872) Using this condition and 119878(119879119872) Killing it iseasy to show that

(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)

which implies that all ℎℓ119894vanish Therefore nabla on119872 is a Levi-

Civita connection For this case coisotropic 119872 will be totallygeodesic in 119872

Finally we quote a general result on the existence of aunique metric connection

Theorem 34 (see [6]) Let119872 be an 119903-lightlike or a coisotropicsubmanifold of a semi-Riemannian manifold Then theinduced linear connection nabla on 119872 is a unique metric connec-tion if and only if one of the following holds

(a) 119860lowast


(b) Rad TM is a Killing distribution

(c) Rad TM is a parallel distribution with respect to nabla

On an induced symmetric Ricci tensor of119872 we quote thefollowing two results

Theorem 35 (see [24]) Let (119872 119892 119878(119879119872)) be an 119903-lightlikeor a coisotropic submanifold of a semi-Riemannian manifoldThen the induced Ricci tensor on119872 is symmetric if and only ifeach Trace(120591

119894119895) induced by 119878(119879119872) is closed where the 1-forms

120591119894119895are as given in (75)

Theorem 36 Let (119872 119892 119878(119879119872perp)) be a screen conformal 119903-lightlike or a coisotropic submanifold of an indefinite space form(119872(119888) 119892) such that 119878(119879119872perp) is Killing Then 119872 admits aninduced symmetric Ricci tensor

Proof Consider a class of screen conformal lightlike subman-ifolds (119872 119892 119878(119879119872) 119878(119879119872

perp)) with Killing 119878(119879119872perp) Using(76) and (81) and taking the Lie derivative pound of 119892 with respectto any base vector 119882

120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry

Using the above relation Killing 119878(119879119872perp) the conformalscreen relation (91) and some curvature properties of thespace form119872(119888) it is easy to prove that119872 admits an inducedsymmetric Ricci tensor which completes the proof

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author thanks the referee sincerely for suggestionstowards the improvement of this revised version The authoris grateful to Professors D H Jin C Atindogbe and B Sahinfor their collaborative work in proving the unique existencetheorems presented in this paper Also thanks are extendedto all authors of the books and articles whose work has beenquoted in this paper

References

[1] B-Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[2] J K Beem and P E Ehrlich Global Lorentzian Geometry vol67 Marcel Dekker New York NY USA 2nd edition 1981

[3] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 Academic Press New York NY USA 1983

[4] M Berger Riemannian Geometry During the Second Half of theTwentieth Century vol 17 ofUniversity Lecture Series AmericanMathematical Society 2000

[5] D N Kupeli Singular Semi-Riemannian Geometry vol 366Kluwer Academic 1996

[6] K L Duggal and A Bejancu Lightlike submanifolds of semi-Riemannian manifolds and applications vol 364 Kluwer Aca-demic Publishers Dordrecht The Netherlands 1996

[7] K L Duggal and D H Jin Null Curves and Hypersurfacesof Semi-Riemannian Manifolds World Scientific PublishingHackensack NJ USA 2007

[8] K L Duggal and B Sahin Differential geometry of lightlikesubmanifolds Frontiers in Mathematics Birkhauser BaselSwitzerland 2010

[9] A Bejancu ldquoLightlike curves in Lorentz manifoldsrdquo Publica-tionesMathematicaeDebrecen vol 44 no 1-2 pp 145ndash155 1994

[10] A Bejancu and K L Duggal ldquoDegenerated hypersurfacesof semi-Riemannian manifoldsrdquo Buletinul Institului PolitehnicIasi vol 37 pp 13ndash22 1991

[11] K L Duggal ldquoA report on canonical null curves and screendistributions for lightlike geometryrdquo Acta Applicandae Mathe-maticae vol 95 no 2 pp 135ndash149 2007

[12] B Sahin ldquoBibliography of lightlike geometryrdquo httpweb2inonuedutrsimlightlike

[13] E Cartan La Theorie des Groupes Finis et Continus et LaGeometrie Differentielle Gauthier-Villars Paris France 1937

[14] W B Bonnor ldquoNull curves in a Minkowski space-timerdquo TheTensor Society vol 20 pp 229ndash242 1969

[15] A Ferrandez A Gimenez and P Lucas ldquoNull helices in Lore-ntzian space formsrdquo International Journal of Modern Physics AParticles and Fields vol 16 no 30 pp 4845ndash4863 2001

[16] R L Ricca ldquoThe contributions of Da Rios and Levi-Civitato asymptotic potential theory and vortex filament dynamicsrdquoJapan Society of Fluid Mechanics Fluid Dynamics Research vol18 no 5 pp 245ndash268 1996

[17] P Lucas ldquoCollection of papers (with his collaborators) onnull curves and applications to physicsrdquo httpwwwumesdocenciaplucas

[18] L P EisenhartATreatise onDifferential Geometry of Curves andSurfaces Ginn and Company 1909

[19] K Honda and J-i Inoguchi ldquoDeformations of Cartan framednull curves preserving the torsionrdquo Differential GeometryDynamical Systems vol 5 no 1 pp 31ndash37 2003

[20] J-i Inoguchi and S Lee ldquoNull curves in Minkowski 3-spacerdquoInternational Electronic Journal of Geometry vol 1 no 2 pp 40ndash83 2008

[21] A C Coken and U Ciftci ldquoOn the Cartan curvatures of a nullcurve in Minkowski spacetimerdquo Geometriae Dedicata vol 114pp 71ndash78 2005

[22] G Mehmet and K Sadik ldquo(1 2)-null Bertrand curves inMinkowski spacetimerdquo vol 1 12 pages 2011

[23] M Sakaki ldquoNotes on null curves inMinkowski spacesrdquo TurkishJournal of Mathematics vol 34 no 3 pp 417ndash424 2010

[24] K L Duggal and D H Jin ldquoGeometry of null curvesrdquo Mathe-matics Journal of Toyama University vol 22 pp 95ndash120 1999

[25] A Ferrandez A Gimenez and P Lucas ldquoDegenerate curvesin pseudo-Euclidean spaces of index twordquo in Proceedings ofthe 3rd International Conference on Geometry Integrability andQuantization pp 209ndash223 Coral Press Sofia Bulgaria 2001

[26] M A Akivis and V V Goldberg ldquoThe geometry of lightlikehypersurfaces of the de Sitter spacerdquo Acta Applicandae Math-ematicae vol 53 no 3 pp 297ndash328 1998

[27] M A Akivis and V V Goldberg ldquoLightlike hypersurfaces onmanifolds endowed with a conformal structure of Lorentziansignaturerdquo Acta Applicandae Mathematicae vol 57 no 3 pp255ndash285 1999

[28] M A Akivis and V V Goldberg ldquoOn some methods of con-struction of invariant normalizations of lightlike hypersur-facesrdquo Differential Geometry and Its Applications vol 12 no 2pp 121ndash143 2000

[29] W B Bonnor ldquoNull hypersurfaces in Minkowski space-timerdquoThe Tensor Society vol 24 pp 329ndash345 1972

[30] T Leistner ldquoScreen bundles of Lorentzian manifolds and somegeneralisations of pp-wavesrdquo Journal of Geometry and Physicsvol 56 no 10 pp 2117ndash2134 2006

[31] V J Bolos ldquoGeometric description of lightlike foliations byan observer in general relativityrdquo Mathematical Proceedings ofthe Cambridge Philosophical Society vol 139 no 1 pp 181ndash1922005

[32] A Bejancu ldquoA canonical screen distribution on a degeneratehypersurfacerdquo Scientific Bulletin A Applied Mathematics andPhysics vol 55 pp 55ndash61 1993

[33] A Bejancu A Ferrandez and P Lucas ldquoA new viewpoint ongeometry of a lightlike hypersurface in a semi-Euclidean spacerdquoSaitama Mathematical Journal vol 16 pp 31ndash38 1998

[34] C Atindogbe and K L Duggal ldquoConformal screen on lightlikehypersurfacesrdquo International Journal of Pure and Applied Math-ematics vol 11 no 4 pp 421ndash442 2004

[35] N Nomizu and S SasakiAfineDifferential Geometry Geometryof Afine Immersions Cambridge University Press CambridgeUK 1994

Geometry 17

[36] K L Duggal ldquoOn scalar curvature in lightlike geometryrdquoJournal of Geometry and Physics vol 57 no 2 pp 473ndash481 2007

[37] C Atindogbe ldquoScalar curvature on lightlike hypersurfacesrdquoApplied Sciences vol 11 pp 9ndash18 2009

[38] C Atindogbe O Lungiambudila and J Tossa ldquoScalar curvatureand symmetry properties of lightlike submanifoldsrdquo TurkishJournal of Mathematics vol 37 no 1 pp 95ndash113 2013

[39] K L Duggal ldquoOn canonical screen for lightlike submanifoldsof codimension twordquo Central European Journal of Mathematicsvol 5 no 4 pp 710ndash719 2007

[40] K L Duggal ldquoOn existence of canonical screens for coisotropicsubmanifoldsrdquo International Electronic Journal of Geometry vol1 no 1 pp 25ndash32 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Geometry

how to deal with this nonuniqueness problem for null curvesand hypersurfaces [11] Since then considerable further workhas been done on these issues in particular reference to alltypes of null curves and submanifolds which has providedstrong foundation for the lightlike geometry

The objective of this second report is to review up-to-dateresults on canonical or unique existence of all types of nullcurves and screen distributions and then find those lightlikesubmanifolds which also admit a uniquemetric connection asymmetric Ricci tensor andhow to recover the induced scalarcurvature subject to some reasonable geometric conditionsWe also propose open problems Our approach is to give briefinformation on themotivation for dealingwith each anomalychronological development of themain results and a sketch oftheir proofswith examples In order to include a large numberof results in one paper we provide a good bibliography withthe aim to encourage those wishing to pursue this subjectfurther More details on these and related works may be seeninBibliography of LightlikeGeometry prepared by Sahin [12]

2 Canonical or Unique NongeodesicNull Curves

Let119862 be a smooth curve immersed in an (119898+2)-dimensionalproper semi-Riemannian manifold (119872 = 119872119898+2

119902 119892) of a

constant index 119902 ge 1 By proper we mean that 119902 is nonzeroWith respect to a local coordinate neighborhoodU on 119862 anda parameter 119905 119862 is given by

119909119894

= 119909119894

(119905) 119894 isin 0 119898 + 1

rank (1198891199090

119905sdot sdot sdot 119889119909

119898+1

119905) = 1 forall119905 isin 119868

(2)

where 119868 is an open interval of a real line and we denote each119889119909119894119889119905 by 119889119909119894

119905 The nonzero tangent vector field onU is given

by 119889119905equiv (1198891199090

119905 119889119909119898+1

119905) equiv 120585

Suppose the curve 119862 is a null curve which preserves itscausal character Then all its tangent vectors are null Thus119862 is a null curve if and only if at each point 119909 of 119862 we have119892(120585 120585) = 0 The normal bundle of 119879119862 is given by

119879119862perp

= 119883 isin Γ (119879119872) 119892 (119883 120585) = 0

dim (119879119862perp

)119909= 119898 + 1

(3)

However null curves behave differently compared to thenonnull curves as follows

(1) 119879119862perp is also a null bundle subspace of 119879119872(2) 119879119862 cap 119879119862perp = 119879119862 rarr 119879119862 oplus 119879119862perp = 119879119872

Thus contrary to the case of nonnull curves since thenormal bundle 119879119862perp contains the tangent bundle 119879119862 of 119862the sum of these two bundles is not the whole of the tangentbundle 119879119872 In other words a vector of 119879

119909119872 cannot be

decomposed uniquely into a component tangent to 119862 and acomponent perpendicular to119862 Moreover since the length ofany arc of a null curve is zero arc-length parameter makes nosense for null curves For these reasons in general a Frenet

frame (constructed by Bejancu [9] in 1994) on a Lorentzianmanifold 119872 along a null curve 119862 depends on the choiceof a pseudoparameter on 119862 and a complementary (but notorthogonal) vector bundle 119878(119879119862

perp) to 119862 in 119879119862perp calling itsscreen distribution In the following we review how onecan generate a canonical or unique set of Frenet equationssubject to reasonable geometric conditions We discuss thisin two subsections of null curves in Lorentzian and semi-Riemannian manifolds (of index 119902 gt 1) respectively

21 Null Curves in Lorentzian Manifolds (119902 = 1) The mainidea (first used by Cartan [13] in 1937 followed by Bonnor[14] in 1969) is to choose minimum number of curvaturefunctions in the Frenet equations We need the following twogeometric conditions on a curve 119862(119901)

(a) 119862(119901) is nongeodesic with respect to a pseudo-arc-parameter 119901

(b) choose119862(119901) such that its first curvature function is ofunit length

Let 119862(119901) be a nongeodesic null curve of a Minkowskispacetime (119872 = R3

1 119892) with a Frenet frame 120585119873119882 where

119901 is a pseudo-arc-parameter and

119892 (120585119873) = 1 = 119892 (119882119882) 119892 (119873119873) = 119892 (120585 120585) = 0

(4)

To deal with the problem of nonuniqueness Cartan con-structed the following called Cartan Frenet frame for 119862(119901)

by using the above two conditions

nabla120585120585 = 119882 nabla

120585119873 = 120591119882 nabla

120585119882 = minus120591120585 minus 119873 (5)

where the torsion function 120591 is invariant up to a sign underLorentzian transformations The above frame is now calledthe null Cartan frame and the corresponding curve is the nullCartan curve (also see Bonnor [14] for the 4-dimensionalcase using Cartan method) In 1994 Bejancu [9] proved thefollowing fundamental existence and uniqueness theoremfor null curves of an (119898 + 2)-dimensional Minkowski space(119877

119898+2

1 119892) First we define in 119877119898+2

1a quasiorthonormal basis

119882119900= (minus

1

radic2

1

radic2 0 0)

1= (

1

radic2

1

radic2 0 0)

2= (0 0 1 0 0)

119898+1

= (0 0 0 1)

(6)

where 119900

1 are null vectors such that 119892(

119900

1) = 1 and

2

119898+1 are orthonormal spacelike vectors It is easy

to see that

119894

119900

119895

1+

119895

119900

119894

1+

119898+1

sum120572=2

119894

120572

119895

120572= ℎ

119894119895 (7)

Geometry 3


ℎ119894119895

=

minus1 119894 = 119895 = 0

1 119894 = 119895 = 0

0 119894 = 119895

(8)

Theorem 1 (see [9]) Let 1198961 119896

2119898 [minus120598 120598] rarr 119877 be



of 119877119898+2

1 let and

119900



1given by


= 119909119894


119894

119900= 119909

119894

(0) and 1198961 119896

2119898are


2 119882


119889

119889119901(0) =

0 119873 (0) =

1

119882120572(0) =

120572 120572 isin 2 119898 + 1

(9)



1198821015840

119900(119901) = 119896

1119882

2

1198821015840

1(119901) = 119896

2119882

2+ 119896

3119882

3

1198821015840

2(119901) = minus 119896

2119882

119900minus 119896

1119882

1+ 119896

4119882

3+ 119896

5119882

4

1198821015840

3(119901) = minus 119896

3119882

119900minus 119896

4119882

2+ 119896

6119882

4+ 119896

7119882

5

1198821015840

119898minus1(119901) = minus 119896

2119898minus5119882


2119898minus4119882

119898minus2

+ 1198962119898minus3

119882119898

+ 1198962119898minus1

119882119898+1

1198821015840

119898(119901) = minus 119896

2119898minus4119882


2119898minus2119882

119898minus1+ 119896

2119898119882

119898+1

1198821015840

119898+1(119901) = 119896

2119898minus1119882


2119898119882

119898

(10)


119898+1 satisfying


119894 119894 isin 0 119898 + 1

Furthermore 119882119900(119901) 119882


basis such that 119882119900(119901)119882

1(119901) and 119882

2(119901) 119882

119898+1(119901) are


119865 = 119889

119889119901= 119882

119900(119901) 119873 (119901) = 119882

1(119901)

1198822(119901) 119882

119898+1(119901)

(11)


2119898







1 119882

119898 satisfying

nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus120581

1120585 minus 119873

nabla120585119882

2= minus120581

2120585 + 120581

3119882

3

nabla120585119882

119894= minus120581

119894119882

119894minus1+ 120581

119894+1119882

119894+1 119894 isin 3 119898 minus 1

nabla120585119882

119898= minus120581

119898119882

119898minus1

(12)


(i) 1198621015840(119901) 11986210158401015840(119901) 119862(119894+2)(119901) and 1205851198731198821 119882

119894


(ii) 1205851198731198821 119882


119894gt 0 for

all 119894 ge 2



1are





120585 =1


119873 = minus1


1198821=

1


1198822=

1


1198823= minus

1


(13)

4 Geometry


nabla120585120585 = 119882

1 nabla

120585119873 = minus

1

6119882

1minus

4

3radic2119882

2

nabla120585119882

1=

1

6120585 minus 119873 nabla

120585119882

2=

4

3radic2120585 +

radic2

3119882

3

nabla120585119882

3= minus

radic2

3119882

2

(14)



1



1and 119882

1are linearly





1



119865 = 1198621015840

119873 =1



(15)




1case


1using the


nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus 120581

1120585 minus 119873

nabla120585119882

2= minus 120581

2120585

(16)



terization theorem


1is a Bertrand


2is zero





1



119898+2



1 119882

119898



2 any


1198651= 120585119873119882

1 119882

119898 (17)




nabla120585120585 = ℎ120585 + 120581

1119882

1

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2

1205981nabla120585119882

1= minus 120581

2120585 minus 120581

1119873 + 120581

4119882

2+ 120581

5119882

3

Geometry 5

1205982nabla120585119882

2= minus 120581

3120585 minus 120581

4119882

1+ 120581

6119882

3+ 120581

7119882

4

1205983nabla120585119882

3= minus 120581

5119882

1minus 120581

6119882

2+ 120581

8119882

4+ 120581

9119882

5

120598119898minus1

nabla120585119882

119898minus1= minus 120581

2119898minus3119882

119898minus3


119882119898minus2

+ 1205812119898

119882119898

120598119898nabla120585119882

119898= minus 120581

2119898minus1119882


2119898119882

119898minus1

(18)

where ℎ and 1205811 120581


1198821 119882


perp

)|U) and

(1205981 120598


119898+2

2such that

120598119894120575119894119895

= 119892(119882119894119882


1 120581

2119898 are called



2given by

119862 (cosh 1199051

271199053

+ 21199051

31199052

1

271199053

radic3119905 sinh 119905) 119905 isin R

(19)


1205851198731198821119882

2119882

3119882

4

120585 = (sinh 1199051199052

9+ 2

2

3119905

1199052


119873 =1

2(sinh 119905 minus

1199052

9minus 2 minus

2

3119905 minus

1199052


1198821= minus

3

radic5(cosh 119905

2

9119905

2

32

9119905 0 sinh 119905)

1198822= minus

1

radic5(2 cosh 119905 119905 3 119905 0 2 sinh 119905)

1198823= minus(0

1199052

9+

3

22

3119905

1199052

9minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(20)



radic5

3119882

1 nabla

120585119873 = minus

13

6radic5119882

1+

2

radic5119882

2

1205981nabla120585119882

1=

13

6radic5120585 +

radic5

3119873 +

4

3radic5119882

3

1205982nabla120585119882

2= minus

2

radic5120585 minus

2

radic5119882

3

1205983nabla120585119882

3= minus

4

3radic5119882

1+

2

radic5119882

2

1205983nabla120585119882

4= 0 ℎ = 0

1205811= minus

radic5

3 120581

2= minus

13

6radic5

1205813=

2

radic5 120581

4= 0 120581

5=

4

3radic5

1205816= minus

2

radic5 120581

7= 0 120581

8= 0

(21)


119894and 119871

119894+1such that

119871119894=

119882119894+ 119882

119894+1

radic2 119871

119894+1=

119882119894+1

minus 119882119894

radic2

119892 (119871119894 119871

119894+1) = 1

(22)

where119882119894and119882



120572 of 119865

1has



1198652= 120585119873119882

1 119871

119894 119871

119894+1 119882

119898 (23)



nabla120585120585 = ℎ120585 + 120581

1119882

1+ 120591

1119882

2

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2+ 120591

2119882

3

nabla120585119882

1= 120581

2120585 + 120581

1119873 minus 120581

4119882

2minus 120581

5119882

3minus 120591

3119882

4

nabla120585119882

2= minus120581

3120585 minus 120591

1119873 minus 120581

4119882

1

+ 1205816119882

3+ 120581

7119882

4+ 120591

4119882

5

nabla120585119882

3= minus120591

2120585 minus 120581

5119882

1minus 120581

6119882

2

+ 1205818119882

4+ 120581

9119882

5+ 120591

5119882

6

nabla120585119882

4= minus120591

3119882

1minus 120581

7119882

2minus 120581

8119882

3

+ 12058110119882

5+ 120581

11119882

6+ 120591

6119882

7

nabla120585119882


119898minus2119882


2119898minus3119882

119898minus3


119882119898minus2

+ 1198962119898

119882119898

nabla120585119882

119898= minus120591

119898minus1119882


2119898minus1119882

119898minus2

minus 1198962119898

119882119898minus1

(24)

6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)


1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052


1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1


119905

2 1

119905


1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)


2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712


1

radic2120585 +

1

radic21198711


1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)


2


3 Their


1 119882







2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)


1are


1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)






119902have 119865





if 119902 = 2 then 119872119898+2



119872119898+2





3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)


1 119871

2 119871

3 119871

4 given as


119873 =1



1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)


nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 3


ℎ119894119895

=

minus1 119894 = 119895 = 0

1 119894 = 119895 = 0

0 119894 = 119895

(8)

Theorem 1 (see [9]) Let 1198961 119896

2119898 [minus120598 120598] rarr 119877 be



of 119877119898+2

1 let and

119900



1given by


= 119909119894


119894

119900= 119909

119894

(0) and 1198961 119896

2119898are


2 119882


119889

119889119901(0) =

0 119873 (0) =

1

119882120572(0) =

120572 120572 isin 2 119898 + 1

(9)



1198821015840

119900(119901) = 119896

1119882

2

1198821015840

1(119901) = 119896

2119882

2+ 119896

3119882

3

1198821015840

2(119901) = minus 119896

2119882

119900minus 119896

1119882

1+ 119896

4119882

3+ 119896

5119882

4

1198821015840

3(119901) = minus 119896

3119882

119900minus 119896

4119882

2+ 119896

6119882

4+ 119896

7119882

5

1198821015840

119898minus1(119901) = minus 119896

2119898minus5119882


2119898minus4119882

119898minus2

+ 1198962119898minus3

119882119898

+ 1198962119898minus1

119882119898+1

1198821015840

119898(119901) = minus 119896

2119898minus4119882


2119898minus2119882

119898minus1+ 119896

2119898119882

119898+1

1198821015840

119898+1(119901) = 119896

2119898minus1119882


2119898119882

119898

(10)


119898+1 satisfying


119894 119894 isin 0 119898 + 1

Furthermore 119882119900(119901) 119882


basis such that 119882119900(119901)119882

1(119901) and 119882

2(119901) 119882

119898+1(119901) are


119865 = 119889

119889119901= 119882

119900(119901) 119873 (119901) = 119882

1(119901)

1198822(119901) 119882

119898+1(119901)

(11)


2119898







1 119882

119898 satisfying

nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus120581

1120585 minus 119873

nabla120585119882

2= minus120581

2120585 + 120581

3119882

3

nabla120585119882

119894= minus120581

119894119882

119894minus1+ 120581

119894+1119882

119894+1 119894 isin 3 119898 minus 1

nabla120585119882

119898= minus120581

119898119882

119898minus1

(12)


(i) 1198621015840(119901) 11986210158401015840(119901) 119862(119894+2)(119901) and 1205851198731198821 119882

119894


(ii) 1205851198731198821 119882


119894gt 0 for

all 119894 ge 2



1are





120585 =1


119873 = minus1


1198821=

1


1198822=

1


1198823= minus

1


(13)

4 Geometry


nabla120585120585 = 119882

1 nabla

120585119873 = minus

1

6119882

1minus

4

3radic2119882

2

nabla120585119882

1=

1

6120585 minus 119873 nabla

120585119882

2=

4

3radic2120585 +

radic2

3119882

3

nabla120585119882

3= minus

radic2

3119882

2

(14)



1



1and 119882

1are linearly





1



119865 = 1198621015840

119873 =1



(15)




1case


1using the


nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus 120581

1120585 minus 119873

nabla120585119882

2= minus 120581

2120585

(16)



terization theorem


1is a Bertrand


2is zero





1



119898+2



1 119882

119898



2 any


1198651= 120585119873119882

1 119882

119898 (17)




nabla120585120585 = ℎ120585 + 120581

1119882

1

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2

1205981nabla120585119882

1= minus 120581

2120585 minus 120581

1119873 + 120581

4119882

2+ 120581

5119882

3

Geometry 5

1205982nabla120585119882

2= minus 120581

3120585 minus 120581

4119882

1+ 120581

6119882

3+ 120581

7119882

4

1205983nabla120585119882

3= minus 120581

5119882

1minus 120581

6119882

2+ 120581

8119882

4+ 120581

9119882

5

120598119898minus1

nabla120585119882

119898minus1= minus 120581

2119898minus3119882

119898minus3


119882119898minus2

+ 1205812119898

119882119898

120598119898nabla120585119882

119898= minus 120581

2119898minus1119882


2119898119882

119898minus1

(18)

where ℎ and 1205811 120581


1198821 119882


perp

)|U) and

(1205981 120598


119898+2

2such that

120598119894120575119894119895

= 119892(119882119894119882


1 120581

2119898 are called



2given by

119862 (cosh 1199051

271199053

+ 21199051

31199052

1

271199053

radic3119905 sinh 119905) 119905 isin R

(19)


1205851198731198821119882

2119882

3119882

4

120585 = (sinh 1199051199052

9+ 2

2

3119905

1199052


119873 =1

2(sinh 119905 minus

1199052

9minus 2 minus

2

3119905 minus

1199052


1198821= minus

3

radic5(cosh 119905

2

9119905

2

32

9119905 0 sinh 119905)

1198822= minus

1

radic5(2 cosh 119905 119905 3 119905 0 2 sinh 119905)

1198823= minus(0

1199052

9+

3

22

3119905

1199052

9minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(20)



radic5

3119882

1 nabla

120585119873 = minus

13

6radic5119882

1+

2

radic5119882

2

1205981nabla120585119882

1=

13

6radic5120585 +

radic5

3119873 +

4

3radic5119882

3

1205982nabla120585119882

2= minus

2

radic5120585 minus

2

radic5119882

3

1205983nabla120585119882

3= minus

4

3radic5119882

1+

2

radic5119882

2

1205983nabla120585119882

4= 0 ℎ = 0

1205811= minus

radic5

3 120581

2= minus

13

6radic5

1205813=

2

radic5 120581

4= 0 120581

5=

4

3radic5

1205816= minus

2

radic5 120581

7= 0 120581

8= 0

(21)


119894and 119871

119894+1such that

119871119894=

119882119894+ 119882

119894+1

radic2 119871

119894+1=

119882119894+1

minus 119882119894

radic2

119892 (119871119894 119871

119894+1) = 1

(22)

where119882119894and119882



120572 of 119865

1has



1198652= 120585119873119882

1 119871

119894 119871

119894+1 119882

119898 (23)



nabla120585120585 = ℎ120585 + 120581

1119882

1+ 120591

1119882

2

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2+ 120591

2119882

3

nabla120585119882

1= 120581

2120585 + 120581

1119873 minus 120581

4119882

2minus 120581

5119882

3minus 120591

3119882

4

nabla120585119882

2= minus120581

3120585 minus 120591

1119873 minus 120581

4119882

1

+ 1205816119882

3+ 120581

7119882

4+ 120591

4119882

5

nabla120585119882

3= minus120591

2120585 minus 120581

5119882

1minus 120581

6119882

2

+ 1205818119882

4+ 120581

9119882

5+ 120591

5119882

6

nabla120585119882

4= minus120591

3119882

1minus 120581

7119882

2minus 120581

8119882

3

+ 12058110119882

5+ 120581

11119882

6+ 120591

6119882

7

nabla120585119882


119898minus2119882


2119898minus3119882

119898minus3


119882119898minus2

+ 1198962119898

119882119898

nabla120585119882

119898= minus120591

119898minus1119882


2119898minus1119882

119898minus2

minus 1198962119898

119882119898minus1

(24)

6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)


1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052


1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1


119905

2 1

119905


1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)


2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712


1

radic2120585 +

1

radic21198711


1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)


2


3 Their


1 119882







2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)


1are


1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)






119902have 119865





if 119902 = 2 then 119872119898+2



119872119898+2





3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)


1 119871

2 119871

3 119871

4 given as


119873 =1



1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)


nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Geometry


nabla120585120585 = 119882

1 nabla

120585119873 = minus

1

6119882

1minus

4

3radic2119882

2

nabla120585119882

1=

1

6120585 minus 119873 nabla

120585119882

2=

4

3radic2120585 +

radic2

3119882

3

nabla120585119882

3= minus

radic2

3119882

2

(14)



1



1and 119882

1are linearly





1



119865 = 1198621015840

119873 =1



(15)




1case


1using the


nabla120585120585 = 119882

1

nabla120585119873 = 120581

1119882

1+ 120581

2119882

2

nabla120585119882

1= minus 120581

1120585 minus 119873

nabla120585119882

2= minus 120581

2120585

(16)



terization theorem


1is a Bertrand


2is zero





1



119898+2



1 119882

119898



2 any


1198651= 120585119873119882

1 119882

119898 (17)




nabla120585120585 = ℎ120585 + 120581

1119882

1

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2

1205981nabla120585119882

1= minus 120581

2120585 minus 120581

1119873 + 120581

4119882

2+ 120581

5119882

3

Geometry 5

1205982nabla120585119882

2= minus 120581

3120585 minus 120581

4119882

1+ 120581

6119882

3+ 120581

7119882

4

1205983nabla120585119882

3= minus 120581

5119882

1minus 120581

6119882

2+ 120581

8119882

4+ 120581

9119882

5

120598119898minus1

nabla120585119882

119898minus1= minus 120581

2119898minus3119882

119898minus3


119882119898minus2

+ 1205812119898

119882119898

120598119898nabla120585119882

119898= minus 120581

2119898minus1119882


2119898119882

119898minus1

(18)

where ℎ and 1205811 120581


1198821 119882


perp

)|U) and

(1205981 120598


119898+2

2such that

120598119894120575119894119895

= 119892(119882119894119882


1 120581

2119898 are called



2given by

119862 (cosh 1199051

271199053

+ 21199051

31199052

1

271199053

radic3119905 sinh 119905) 119905 isin R

(19)


1205851198731198821119882

2119882

3119882

4

120585 = (sinh 1199051199052

9+ 2

2

3119905

1199052


119873 =1

2(sinh 119905 minus

1199052

9minus 2 minus

2

3119905 minus

1199052


1198821= minus

3

radic5(cosh 119905

2

9119905

2

32

9119905 0 sinh 119905)

1198822= minus

1

radic5(2 cosh 119905 119905 3 119905 0 2 sinh 119905)

1198823= minus(0

1199052

9+

3

22

3119905

1199052

9minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(20)



radic5

3119882

1 nabla

120585119873 = minus

13

6radic5119882

1+

2

radic5119882

2

1205981nabla120585119882

1=

13

6radic5120585 +

radic5

3119873 +

4

3radic5119882

3

1205982nabla120585119882

2= minus

2

radic5120585 minus

2

radic5119882

3

1205983nabla120585119882

3= minus

4

3radic5119882

1+

2

radic5119882

2

1205983nabla120585119882

4= 0 ℎ = 0

1205811= minus

radic5

3 120581

2= minus

13

6radic5

1205813=

2

radic5 120581

4= 0 120581

5=

4

3radic5

1205816= minus

2

radic5 120581

7= 0 120581

8= 0

(21)


119894and 119871

119894+1such that

119871119894=

119882119894+ 119882

119894+1

radic2 119871

119894+1=

119882119894+1

minus 119882119894

radic2

119892 (119871119894 119871

119894+1) = 1

(22)

where119882119894and119882



120572 of 119865

1has



1198652= 120585119873119882

1 119871

119894 119871

119894+1 119882

119898 (23)



nabla120585120585 = ℎ120585 + 120581

1119882

1+ 120591

1119882

2

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2+ 120591

2119882

3

nabla120585119882

1= 120581

2120585 + 120581

1119873 minus 120581

4119882

2minus 120581

5119882

3minus 120591

3119882

4

nabla120585119882

2= minus120581

3120585 minus 120591

1119873 minus 120581

4119882

1

+ 1205816119882

3+ 120581

7119882

4+ 120591

4119882

5

nabla120585119882

3= minus120591

2120585 minus 120581

5119882

1minus 120581

6119882

2

+ 1205818119882

4+ 120581

9119882

5+ 120591

5119882

6

nabla120585119882

4= minus120591

3119882

1minus 120581

7119882

2minus 120581

8119882

3

+ 12058110119882

5+ 120581

11119882

6+ 120591

6119882

7

nabla120585119882


119898minus2119882


2119898minus3119882

119898minus3


119882119898minus2

+ 1198962119898

119882119898

nabla120585119882

119898= minus120591

119898minus1119882


2119898minus1119882

119898minus2

minus 1198962119898

119882119898minus1

(24)

6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)


1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052


1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1


119905

2 1

119905


1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)


2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712


1

radic2120585 +

1

radic21198711


1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)


2


3 Their


1 119882







2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)


1are


1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)






119902have 119865





if 119902 = 2 then 119872119898+2



119872119898+2





3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)


1 119871

2 119871

3 119871

4 given as


119873 =1



1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)


nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 5

1205982nabla120585119882

2= minus 120581

3120585 minus 120581

4119882

1+ 120581

6119882

3+ 120581

7119882

4

1205983nabla120585119882

3= minus 120581

5119882

1minus 120581

6119882

2+ 120581

8119882

4+ 120581

9119882

5

120598119898minus1

nabla120585119882

119898minus1= minus 120581

2119898minus3119882

119898minus3


119882119898minus2

+ 1205812119898

119882119898

120598119898nabla120585119882

119898= minus 120581

2119898minus1119882


2119898119882

119898minus1

(18)

where ℎ and 1205811 120581


1198821 119882


perp

)|U) and

(1205981 120598


119898+2

2such that

120598119894120575119894119895

= 119892(119882119894119882


1 120581

2119898 are called



2given by

119862 (cosh 1199051

271199053

+ 21199051

31199052

1

271199053

radic3119905 sinh 119905) 119905 isin R

(19)


1205851198731198821119882

2119882

3119882

4

120585 = (sinh 1199051199052

9+ 2

2

3119905

1199052


119873 =1

2(sinh 119905 minus

1199052

9minus 2 minus

2

3119905 minus

1199052


1198821= minus

3

radic5(cosh 119905

2

9119905

2

32

9119905 0 sinh 119905)

1198822= minus

1

radic5(2 cosh 119905 119905 3 119905 0 2 sinh 119905)

1198823= minus(0

1199052

9+

3

22

3119905

1199052

9minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(20)



radic5

3119882

1 nabla

120585119873 = minus

13

6radic5119882

1+

2

radic5119882

2

1205981nabla120585119882

1=

13

6radic5120585 +

radic5

3119873 +

4

3radic5119882

3

1205982nabla120585119882

2= minus

2

radic5120585 minus

2

radic5119882

3

1205983nabla120585119882

3= minus

4

3radic5119882

1+

2

radic5119882

2

1205983nabla120585119882

4= 0 ℎ = 0

1205811= minus

radic5

3 120581

2= minus

13

6radic5

1205813=

2

radic5 120581

4= 0 120581

5=

4

3radic5

1205816= minus

2

radic5 120581

7= 0 120581

8= 0

(21)


119894and 119871

119894+1such that

119871119894=

119882119894+ 119882

119894+1

radic2 119871

119894+1=

119882119894+1

minus 119882119894

radic2

119892 (119871119894 119871

119894+1) = 1

(22)

where119882119894and119882



120572 of 119865

1has



1198652= 120585119873119882

1 119871

119894 119871

119894+1 119882

119898 (23)



nabla120585120585 = ℎ120585 + 120581

1119882

1+ 120591

1119882

2

nabla120585119873 = minus ℎ119873 + 120581

2119882

1+ 120581

3119882

2+ 120591

2119882

3

nabla120585119882

1= 120581

2120585 + 120581

1119873 minus 120581

4119882

2minus 120581

5119882

3minus 120591

3119882

4

nabla120585119882

2= minus120581

3120585 minus 120591

1119873 minus 120581

4119882

1

+ 1205816119882

3+ 120581

7119882

4+ 120591

4119882

5

nabla120585119882

3= minus120591

2120585 minus 120581

5119882

1minus 120581

6119882

2

+ 1205818119882

4+ 120581

9119882

5+ 120591

5119882

6

nabla120585119882

4= minus120591

3119882

1minus 120581

7119882

2minus 120581

8119882

3

+ 12058110119882

5+ 120581

11119882

6+ 120591

6119882

7

nabla120585119882


119898minus2119882


2119898minus3119882

119898minus3


119882119898minus2

+ 1198962119898

119882119898

nabla120585119882

119898= minus120591

119898minus1119882


2119898minus1119882

119898minus2

minus 1198962119898

119882119898minus1

(24)

6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)


1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052


1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1


119905

2 1

119905


1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)


2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712


1

radic2120585 +

1

radic21198711


1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)


2


3 Their


1 119882







2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)


1are


1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)






119902have 119865





if 119902 = 2 then 119872119898+2



119872119898+2





3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)


1 119871

2 119871

3 119871

4 given as


119873 =1



1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)


nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 Geometry


2given by

119862 (cosh 1199051

121199053

+ 21199051

21199052

1

121199053

radic3119905 sinh 119905) 119905 isin 119877

(25)


1 119871

2119882

3119882

4 of Type 2

as follows

120585 = (sinh 1199051199052

4+ 2 119905

1199052


119873 =1

2(sinh 119905 minus

1199052


1199052


1198711=

1

radic2(cosh 119905

119905

2 1

119905

2 0 sinh 119905)

1198712=

1


119905

2 1

119905


1198823= minus(0

1199052

4+

3

2 119905

1199052

4minus

1

2radic3 0)

1198824= (0

radic3

2 0

radic3

2 1 0)

(26)


2

nabla120585120585 = radic2119871

1 ℎ = 0 nabla

120585119873 = minus

1

radic21198712


1

radic2120585 +

1

radic21198711


1

radic2119882

3

nabla120585119882

3= minus

1

radic21198711minus

1

radic21198712

nabla120585119882

4= 0

(27)


2


3 Their


1 119882







2 119871

3 and 119871

4such that

1198711=

1198821+ 119882

2

radic2 119871

2=

1198822minus 119882

1

radic2 119892 (119871

1 119871

2) = 1

1198713=

1198823+ 119882

4

radic2 119871

4=

1198824minus 119882

3

radic2 119892 (119871

3 119871

4) = 1

(28)


1are


1198653= 120585119873 119871

1 119871

2 119871

3 119871

4119882

5 119882

119898 (29)






119902have 119865





if 119902 = 2 then 119872119898+2



119872119898+2





3given by

119862 (cos 119905 sin 119905 sinh 119905 cosh 119905 119905 119905) (30)


1 119871

2 119871

3 119871

4 given as


119873 =1



1198712=

1

2(cos 119905 sin 119905 sinh 119905 cosh 119905 0 0)

1198713= (minus sin 119905 cos 119905 0 0 0 1)

1198714= (0 0 cosh 119905 sinh 119905 0 1)

(31)


nabla120585120585 = 119871

1 nabla

120585119873 = minus

1

21198711

nabla1205851198711= minus119871

3+ 119871

4

nabla1205851198712=

1

2120585 minus 119873 minus

1

2(119871

3+ 119871

4)

nabla1205851198713=

1

21198711minus 119871

2 nabla

1205851198714=

1

21198711

(32)

Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 7


119898+2

119902 119892)


119902 Sup-



119909119860

= 119909119860

(119905) 119905 isin I sub 119877 119860 isin 0 1 (119898 + 1) (33)


119892 (119909 119910) = minus(

119902minus1

sum119894=0

119909119894

119910119894

) + (

119898+1

sum120572=119902

119909120572

119910120572

) (34)

Let 1198651

= 1205851198731198821 119882


of Type 1 Take 1198821 119882





119865119902= 120585 = 119871

1 119873 = 119871

lowast

1 119871

2

119871119902 119871

lowast

2 119871

lowast

119902119882

119902+1 119882

119898

(35)


1198711= (

1

radic2 0 0 0 0 0 0 0

1

radic2)

119871lowast

1= (minus

1

radic2 0 0 0 0 0 0 0

1

radic2)

1198712= (0

1

radic2 0 0 0 0 0

1

radic2 0)

119871lowast

2= (0 minus

1

radic2 0 0 0 0 0

1

radic2 0)

1198713= (0 0

1

radic2 0 0 0 0

1

radic2 0 0)

119871lowast

3= (0 0 minus

1

radic2 0 0 0 0

1

radic2 0 0)

119871119902= (0 0

1

radic2

1

radic2 0 0)

119871lowast

119902= (0 0 minus

1

radic2

1

radic2 0 0)

(36)

where 1198711 119871

119902 119871lowast

1 119871lowast


119892 (119871119894 119871

lowast

119895) = 120575

119894119895 119892 (119871

119894 119871

119895) = 0

119892 (119871lowast

119894 119871

lowast

119895) = 0

(37)


sum119894=1

(119871119860

119894119871lowast119861

119894+ 119871

119861

119894119871lowast119860

119894) = ℎ

119860119861

(38)


ℎ119860119861

=

minus1 119860 = 119861 isin 0 119902 minus 1

1 119860 = 119861 isin 119902 (119898 + 1)

0 119860 = 119861

(39)


119902(which also


Theorem 14 (see [7]) Let 120581119894 120591

119894 120583

119894 ]

119894 120578

119894 [minus120576 120576] rarr 119877


= (119909119894


of 119877119898+2

119902 and 119871

119894 119871lowast

119894119882

120572 1 le 119894 le 119902 119902 + 1 le 120572 le 119898



119877119898+2

119902such that 119909119894 = 119909119894(119905) 119862(0) = 119909

119900 and 120581

119894 120591

119894 120583

119894 ]

119894 120578

119894



120585 =119889

119889119901= 119871

1 119873 (0) = 119871

lowast

1

119882120572(0) = 119882

120572 119902 + 1 le 120572 le 119898

(40)



119898+2








8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 Geometry


119909119872 of 119879

119909119872 at each


Rad119879119909119872 = 120585 isin 119879

119909119872 119892

119909(120585 119883) = 0 = 119879

119909119872

perp

forall119883 isin 119879119909119872

(41)





119909119872perp



119879119872 = Rad119879119872oplusorth119878 (119879119872) (42)


119879119872|119872

= 119878 (119879119872) perp 119878(119879119872)perp

119878 (119879119872) cap 119878(119879119872)perp

= 0

(43)


|119872




119892 (119873 120585) = 1

119892 (119873119873) = 119892 (119873119882) = 0 forall119882 isin Γ (119878119879 (119872) |U) (44)


119906119872 =


119879119872|119872

= 119878 (119879119872)oplusorth (119879119872perp

oplus tr (119879119872)) = 119879119872 oplus tr (119879119872)

(45)





119883119884 + 119861 (119883 119884)119873

nabla119883119873 = minus119860

119873119883 + 120591 (119883)119873

(46)





(nabla119883119892) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884) (47)



nabla119883119875119884 = nabla

lowast

119883119875119884 + 119862 (119883 119875119884) 120585

nabla119883120585 = minus119860

lowast


(48)


119861 (119883 119884) = 119892 (119860lowast

120585119883119884) 119892 (119860

lowast

120585119883119873) = 0

119862 (119883 119875119884) = 119892 (119860119873119883119875119884) 119892 (119860

119873119884119873) = 0

(49)






119902



119902minus1of119877119899+1



Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 9







119886 induced on


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 + f120585 +

119898

sum119886=1

f119886119882

119886

(50)


119886 and119882119887

119886


119886] is 119898 times


119886119882

119886) = 1 we get 2f + sum

119898

119886=1120598119886(f119886)2


1198821015840

119886=

119898

sum119887=1

119882119887

119886(119882

119887minus 120598

119887f119887120585)

1198731015840

= 119873 minus1

2

119898

sum119886=1

120598119886(f

119886)2

120585

+

119898

sum119886=1

f119886119882

119886

(51)






120585of119872 and


119860119873

= 120593119860lowast

120585 (52)



119862 (119883 119875119884) = 120593119861 (119883 119884) forall119883 119884 isin Γ (119879119872|U) (53)


S1

(119909) = span [119883 119884]|119909 119883

119909 119884

119909isin 119878 (119879

119909119872) forall119909 isin 119872

(54)





119898

119886=1f119886119882


existence theorem






lowast







119886= sum

119898

119887=1119882119887

119886119882

119887 (1 le 119886 le 119898) and 1198731015840 = 119873 where

(119882119887


119909119872) at any point 119909 of119872




10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









10 Geometry


119898+2

119902 how-






119898+2

119902 119892) with 119864 a





119872 = (1199090

119909119898+1

) isin 119877119898+2

119902 119909

0

= 119865 (1199091

119909119898+1

)

(55)


is

1199090

= 119865 (V0 V119898) 119909120572+1

= V120572 120572 isin 0 119898

(56)


120597V120572 = 1198651015840

119909120572+1120597

1199090 + 120597

119909120572+1 120572 isin 0 119898 (57)

Then

120585 = 1205971199090 minus

119902minus1

sum119904=1

1198651015840

119909119904120597

119909119904 +

119898+1

sum119886=119902

1198651015840

119909119886120597

119909119886 (58)



1 +

119902minus1

sum119904=1

(1198651015840

119909119904)

2

=

119898+1

sum119886=119902

(1198651015840

119909119886)

2

(59)


Γ(119879119877119898+2






119883119873 120585) = (12)119892(nabla

119883120585 120585) = 0 The Weingarten


119873119883 and nabla

119883120585 = minus119860lowast

120585119883 which

implies

119860119873119883 =

1

2119860lowast

120585119883 forall119883 isin Γ (119879119872) (60)


119902is screen




119883119892 = 0 A distribution 119863











(61)




we obtain

119877 (119883 119884) =

119898

sum119886=1

120598119886119892 (119877 (119883119882

119886) 119884119882

119886) + 119892 (119877 (119883 120585) 119884119873)

(62)

Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 11



119892 (119877 (119883119882119886) 119884119882

119886) = 119892 (119877 (119883119882

119886) 119884119882

119886)

+ 119861 (119883 119884) 119862 (119882119886119882

119886)

minus 119861 (119882119886 119884) 119862 (119883119882

119886)

(63)


119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) 119905119903119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119892 (119877 (120585 119884)119883119873)

forall119883 119884 isin Γ (119879119872)

(64)






119877 (119883 119884) = Ric (119883 119884) + 119861 (119883 119884) tr119860119873

minus 119892 (119860119873119883119860

lowast

120585119884) minus 119888119892 (119883 119884)

(65)



119877 (119883 119884) = Ric (119883 119884) minus 119888119892 (119883 119884) (66)










119903 = 119877 (120585 120585) +

119898

sum119886=1

119877 (119882119886119882

119886)

=

119898

sum119886=1

119898

sum119887=1

119892 (119877 (119882119886119882

119887)119882

119886119882

119887)

+

119898

sum119886=1

119892 (119877 (120585119882119886) 120585119882

119886)

+ 119892 (119877 (119882119886 120585)119882

119886 119873)

(67)







Denote by C[119872]0= [(119872 119892 119878(119879119872))]


12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









12 Geometry






0









of 1198773





11198774

1 and1198774






119879119901119872

perp

= 119881119901isin 119879

119901119872119892

119901(119881

119901119882

119901) = 0 forall119882

119901isin 119879

119901119872

(68)


Rad119879119901119872 = 119879

119901119872 cap 119879

119901119872

perp

= 0 forall119901 isin 119872 (69)







119879119872 = Rad (119879119872)oplusorth119878 (119879119872)

119879119872perp


) (70)


|119872and 119879119872

perp in 119878(119879119872)perp


|U)


|U where U is a


119892 (119873119894 120585

119895) = 120575

119894119895 119892 (119873

119894 119873

119895) = 0 (71)

where 1205851 120585


119879119872 = 119879119872 oplus tr (119879119872)



)

(72)






1205851 120585

119903 119873

1 119873

119903 119883

119903+1 119883

119898119882

119903+1 119882

119899

(73)

where 119883119903+1

119883119898 and 119882

119903+1 119882



Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 13



119883119884 +

119903

sum119894=1

ℎℓ

119894(119883 119884)119873

119894+

119899

sum120572=119903+1

ℎ119904

120572(119883 119884)119882

120572 (74)

nabla119883119873

119894= minus 119860

119873119894

119883 +

119903

sum119895=1

120591119894119895(119883)119873

119895+

119899

sum120572=119903+1

120588119894120572

(119883)119882120572 (75)

nabla119883119882

120572= minus 119860

119882120572

119883 +

119903

sum119894=1

120601120572119894

(119883)119873119894+

119899

sum120573=119903+1

120579120572120573

(119883)119882120573

(76)

nabla119883119875119884 = nabla

lowast

119883119875119884 +

119903

sum119894=1

ℎlowast

119894(119883 119875119884) 120585

119894 (77)

nabla119883120585119894= minus 119860

lowast

120585119894

119883 minus

119903

sum119895=1

120591119895119894(119883) 120585

119895 forall119883 119884 isin Γ (119879119872)

(78)





119894 given in


119873119894

119860119882120572

and119860lowast

120585119894


119894119895 120588

119894120572 120601

120572119894 and 120579



119894and ℎ119904

120572are symmetric


119883119884 120585


119894


119894 120591

119894119895 and 120588


and 119889(tr(120591119894119895)) = 119889(tr(120591



(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884)

(79)


119894(119883) = 119892(119883119873

119894) forall119883 isin Γ(119879119872)




ℎℓ

119894(119883 119884) = 119892 (119860

lowast

120585119894

119883119884) minus

119903

sum119895=1

ℎℓ

119895(119883 120585

119894) 120578

119895(119884)

ℎℓ

119894(119883 119875119884) = 119892 (119860

lowast

120585119894

119883 119875119884) 119892 (119860lowast

120585119894

119883119873119895) = 0

(80)

120598120572ℎ119904

120572(119883 119884) = 119892 (119860

119882120572

119883119884) minus

119903

sum119894=1

120601120572119894

(119883) 120578119894(119884) (81)

120598120572ℎ119904

120572(119883 119875119884) = 119892 (119860

119882120572

119883119875119884)

119892 (119860119882120572

119883119873119894) = 120598

120572120588119894120572

(119883)

(82)

ℎlowast

119894(119883 119875119884) = 119892 (119860

119873119894

119883 119875119884)

120578119895(119860

119873119894

119883) + 120578119894(119860

119873119895

119883) = 0

(83)


119894 119883

119886119882

120572

and 120585119894 119873

1015840

119894 119883

1015840

119886119882

1015840

120572 of 119878(119879119872) 119878(119879119872

perp

) 119865 and 1198781015840

(119879119872)


perp

)perp and 119878

1015840

(119879119872perp

)perp respectively


1198831015840

119886=

119898

sum119887=119903+1

119883119887

119886(119883

119887minus 120598

119887

119903

sum119894=1

f119894119887120585119894)

1198821015840

120572=

119899

sum120573=119903+1

119882120573

120572(119882

120573minus 120598

120573

119903

sum119894=1

119876119894120573120585119894)

1198731015840

119894= 119873

119894+

119903

sum119895=1

119873119894119895120585119895+

119898

sum119886=119903+1

f119894119886119883

119886

+

119899

sum120572=119903+1

119876119894120572119882

120572

(84)

where 120598119886 and 120598


119886 and 119882

120572


119886 119882120573

120572 119873

119894119895 f

119894119886 and 119876


on U such that [119883119887

119886] and [119882120573



119873119894119895+ 119873

119895119894+

119898

sum119886=119903+1

120598119886f119894119886f119895119886

+

119899

sum120572=119903+1

120598120572119876

119894120572119876

119895120572= 0


1015840

119883119884 +

119903

sum119895=1

119903

sum119894=1

ℎ119897

119894(119883 119884)119873

119894119895

minus

119899

sum120572120573=119903+1

120598120573ℎ1015840119904

120572(119883 119884)119882

120573

120572119876

119895120573

120585119895

+

119898

sum119886= 119903+1

119903

sum119894=1

ℎ119897

119894(119883 119884) f

119894119886119883

119886

ℎ119904

120572(119883 119884) =

119903

sum119894=1

ℎ119897

119894(119883 119884)119876

119894120572

+

119899

sum120573=119903+1

ℎ1015840119904

120573(119883 119884)119882

120572

120573

(85)

ℎ119897

119894(119883 119884) = ℎ

1015840119897

119894(119883 119884) forall119883 119884 isin Γ (119879119872) (86)

14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









14 Geometry



ℎ1015840lowast

119894(119883 119875119884) = ℎ

lowast

119894(119883 119875119884) + 119892 (nabla

119883119875119884Z

119894)

minus sum119894 = 119895

ℎ119897

119895(119883 119875119884) [119873

119895119894+ 119892 (Z

119894Z

119895)]

+1

2ℎ119897

119894(119883 119875119884) [

1003817100381710038171003817W119894

10038171003817100381710038172

minus1003817100381710038171003817Z119894

10038171003817100381710038172

]

+

119899

sum120572=119903+1

120576120572ℎ119904

120572(119883 119875119884)119876

119894120572

(87)

where Z119894= sum

119898

119886=119903+1f119894119886119883

119886andW

119894= sum

119899

120572=119903+1119876

119894120572119882

120572


119894given by

120596119894(119883) = 119892 (119883Z



S (119909) = span [119883 119884]119909 119883

119909 119884

119909isin 119878 (119879119872) 119909 isin 119872 (89)


(119883 119884) 119883119909 119884

119909isin 119879

119909119872 that is

N1

(119909) = span ℎ119904

(119883 119884)119909 119883

119909 119884

119909isin 119879

119909119872 (90)


ℓ



lowast








119873119894




ℎlowast



ℎlowast

119894(119883 119875119884) = 120593

119894ℎ119894(119883 119875119884)

forall119883 119884 Γ (119879119872|U) 119894 isin 1 119903 (91)





119898+119899 Then





119903 of Γ(Rad119879119872) with


1 119873





119892 (119860119873119894

119883 119875119884) = 120593119894119892 (119860

120585119894

119883 119875119884) forall119883 isin Γ (119879119872|U)

(92)

Since each 119860120585119894


119873119894



119894119886= 0 Using


119894and all 119876





a basis 119881119894 119894 isin 1 119903 of Γ(119865


119873119894=

119903

sum119896=1

119860119894119896120585119896+ 119861

119894119896119881119896 (93)

where 119860119894119896and 119861


119894

satisfy 119892(119873119894 120585

119894) = 120575

119894119895if and only if sum119903

119896=1119861119894119896119892119895119896

= 120575119894119895 where

Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 15

119892119895119896

= 119892(120585119895 119881


119895119896]








nabla119883119873

119894=

119903

sum119896=1

(119883 (119860119894119896) 120585

119896+ 119860

119894119896nabla119883120585119896

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896)

(94)


119860119873119894

119883 = nabla119897

119883119873

119894+ 119863

119904

(119883119873119894)

minus

119903

sum119896=1

[119883 (119860119894119896) 120585

119896+ 119860

119894119896


120585119896


119883120585119896+ ℎ

119897

(119883 120585119896) + ℎ

119904

(119883 120585119896)

+119883 (119861119894119896) 119881

119896+ 119861

119894119896nabla119883119881119896]

(95)


get 119892(119860119873119894

119883119884) = sum119903

119896=1119860

119894119896119892(119860lowast

120585119896


119892 (ℎlowast

(119883 119884) 119873119894) =

119903

sum119896=1

119860119894119896119892 (ℎ

119897

(119883 119884) 120585119896) (96)






119894








(nabla119883119892) (119884 119885) =

119903

sum119894=1

ℎℓ

119894(119883 119884) 120578

119894(119885) + ℎ

ℓ

119894(119883 119885) 120578

119894(119884) = 0

(97)





(a) 119860lowast







120591119894119895are as given in (75)




120572 we get

(pound119882120572

119892) (119883 119884) = minus2120598120572ℎ119904

120572(119883 119884) (98)

16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









16 Geometry




Acknowledgments


References




































Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 17













Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Review Article A Review on Unique Existence Theorems in ...downloads.hindawi.com/archive/2014/835394.pdf · areas of research in di erential geometry. Most of the work ontheRiemannian,semi-Riemannian,andLorentzianman-ifolds