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Mathematical Computation June 2014, Volume 3, Issue 2, PP.30-37

On Spacelike Hypersurfaces in Anti de Sitter

Four Space Jiajing Miao

School of Science, Mudanjing Normal University, Mudanjing, 157011, P.R. China

#Email: jiajing0407@126.com

Abstract

In this paper, contact geometry of spacelike hypersurfaces in Anti de Sitter four spacewill be studied. To do this, we revealed the

relationship between the singularity Anti de Sitter height function and that of spacelike hypersurfaces. In addition the contact

relations between spacelike hypersurfaces and AdS-great-hyperboloids are studied from geometrical point of view.

Keywords: Anti de Sitter Four Space, Spacelike Hyppersurfaces, Legendrian Singularities

1 INTRODUCTION

Since the second half of the 20th century, the Reimannian and semi-Reimannian geometry have been active areas of

research in differential geometry and its application to variety of subjects in mathematics and physics. During the mid

1970s, the interest shifted towards Lorentzian geometry, the mathematical theory used in general relativity. Since then

there has been amazing leap in the depth of the connection between modern differential geometry and mathematical

relativity, both from the local and the global point of view. A minor revolution in mathematical thought and technique

occurred during 1960s, largely through the inventive genius of French mathematician Rene Thom. His ideas partly

inspired by H. Whitney gave birth to what is called singularity theory, a term which includes catastrophes and

bifurcations. Today's singularity being a direct descendant of differential calculus, is certain to have a great deal of

interests to say about geometry and therefore about all the branches of mathematics, physics and other disciplines where

the geometrical spirit is a guiding light. More recently developments in singularity theory have enriched the field of

geometry by making possible a wealth of detail only dreamed of fifty years ago.

Anti de Sitter space is a maximally symmetric, vacuum solution of the Einstein's field equation with an attractive

cosmological constant in the theory of relativity which makes it be a very important space in both the astrophysics and

geometry [1-3, 8-9]. In this paper, we try to introduce a basic framework for the study of spacelike hypersurfaces in Anti

de Sitter 4-space. This work can be seen as a complement work of those in [2] and some applications of the theory

introduced in [1, 3-6].

The organization of this paper is as follows. In Section 2, we will develop the local differential geometry of spacelike

hypersurfaces in Anti de Sitter4-space. In Section 3, we will study the properties of Anti de Sitter height function and

we prove Anti de Sitter Gauss map to be a wave front set. In Sections 4, we will investigate the contact relations

between spacelike hypersurfaces and AdS-great-hyperboloids. Finally, we will study the generic properties of spacelike

hypersurfac-es.

We assume throughout the whole paper that all the maps and manifolds are C unless the contrary is explicitly stated.

2 BASIC CONCEPTS AND THE LOCAL DIFFERENTIAL GEOMETRY OF SPACELIKE

HYPERSURFACES

In this section, we shall shortly introduce the local differential geometry of spacelike hypersurfaces in Anti de sitter

4-space.

Let 5

1 2 3 4 5, , , , ( 1,2,3,4,5)

iR x x x x x x R i be a 5-dimensional vector space. For any vectors

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1 2 3 4, 5( , , , )x x x x x x and

1 2 3 4 5( , , , , )y y y y y y in 5R , the pseudo scalar product of x and y is defined to be

1 1 2 2 3 3 4 4 5 52,x y x y x y x y x y x y .

We call 5

2( , , )R a semi-Euclidean 5-space with index 2. We write 5

2R instead of 5

2( , , )R . A non-zero vector

5

2x R is called spacelike, null or timelike if

2, 0x x ,

2, 0x x or

2, 0x x respectively. The norm of a vector

5

2x R is defined by

2,x x x .For a vector 5

2n R and a real number c, we define a hyperplane with pseudo-

normal n by

5

2 2( , ) ,HP n c x R x n c

The ( , )HP n c is called a Lorentz hyperplane, a semi-Euclidean hyperplane with index 2 or a null hyperplane if n is

timelike, spacelike or null respectively.

We now define Anti de sitter 4-space (or AdS 4 -space) by

4 5

2 2, 1Ads x R x x ;

Given vectors 5

1 2 3 4 2, , ,X X X X R , we define their wedge product

1 2 3 4X X X X by

1 2 3 4 5

1 2 3 4 5

1 1 1 1 1

1 2 3 4 5

1 2 3 4 2 2 2 2 2

1 2 3 4 5

3 3 3 3 3

1 2 3 4 5

4 4 4 4 4

e e e e e

x x x x x

X X X X x x x x x

x x x x x

x x x x x

Where 1 2 3 4 5, , , ,e e e e e is a canonical basis of 5

2R and 1 2 3 4 5, , , , .

i i i i i iX x x x x x We can easily check that

1 2 3 4 1 2 3 42, det , , , , 0

i iX X X X X X X X X X

So that 1 2 3 4

X X X X is pseudo orthogonal to any 1,2,3,4i

X i .

We now introduce the extrinsic differential geometry of spacelike hypersurfaces in AdS 4 .Consider an embed-ding 4:X U AdS from an open subset 3U R . We write ( )M X U by identifying M with U through the

embedding X . We say that X is spacelike ifiu

X , 1,2,3,i are always spacelike vectors. In this case, we call ( )X u

spacelike hypersurface. We define a vector ( )N u by

1 2 3

1 2 3

( ) ( ) ( ) ( )( )

u u u

u u u

X u X u X u X uN u

X u X u X u X u

By definition, we have

22

, , 0iu

N u X u N u X u

Where 1,2,3i .This indicates that ( )X u , ( )p

N u N M , where 1 2 3, ,u u u u is a local coordinate system

around p X u M . We can easily check that ( )N u is timelike vector and2

, 1N N . Therefore we can define

a map

4:N U Ads by u N u

We call it timelike Anti de sitter Gauss image (or TADS Gauss image) of X (or M ). We now consider the geometric

meaning of the TAdS Gauss image of a spacelike hypersurface. We define typical spacelike surface in AdS 4

by 4( , ) ,AH n c AdS HP n c . We call ,AH n c an AdS-hyperboloid in AdS 4 , if n is timelike and n c .

Especially, we call ,0AH n the AdS-great-hyperboloid if n is timelike. Then we have the following propositions.

Proposition 2.1 Let 4:X U Ads be a spacelike hypersurface in Anti de sitter 4-space. If the TAdS Gauss image

( )N u is constant, then the spacelike hypersurface X u M is a part of an AdS-great- hyperboloid. We can easily

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show thatiu

N , 1,2,3,i are tangent vectors of M . Therefore we have linear transformation

:p p p

W dN u T M T M

Which is called the Anti de Sitter shape operator (or Ads-shape operator) of ( )M X U at p X u . We denote the

eigenvalues of p

W by 1,2,3i

k p i . The Anti de sitter Gauss-Kroneker curvature (or AdS-G-K Curvature) of

( )M X U at p X u is defined by

1 2 3det

AdS pK u W k p k p k p .

We call a point p X u an Anti de Sitter parabolic point (or an AdS-parabolic point) of 4:X U Ads if

0AdS

K u .

We call a point u U or p X u an umbilic point if pp T M

W k p id .Especially, we call ( )M X U totally

umbilic if every point in M is umbilic. Then we can get the following proposition.

Proposition 2.2. Suppose that ( )M X U is totally umbilic. Then k p is constant k . Under this condition we

have the following classification.

(1) If 0k , then M is a part of an AdS-great-hyperboloid 4, 1HP n AdS , where 1

n X Nk

is a constant

timelike vector.

(2) If 0k , then M is a part of an AdS-great-hyperboloid 4,0HP n AdS , where n N is a constant timelike

vector.

In the last part of this section, we introduce the Anti de sitter Weingarten formula. We induce the Anti de Sitter first

fundamental form 3

2

,, 1

i j i ji j

ds g du du

on ( )M X U , where 2

,i jij u u

g u X u X u for any u U .We also

define the Anti de Sitter second fundamental invariant by 2

,i jij u u

h u N u X u for anyu U .

Proposition 2.3 Under the above notations, we have the following Anti de Sitter Weingarten formula:

3

1i j

j

u i uj

N u h u X u

where j kj

i ikh u h u g u and

1kj

kjg u g u

.

As a corollary of the above proposition, we have an explicit expression for the AdS-G-K curvature in terms of the

Riemannian metric and the Anti de Sitter second fundamental invariant.

Corollary 2.4 Under the above notations. The AdS-G-K curvature is given by

det

det

ij

Ads

h uK

g u

We call a point p X u an Anti de Sitter parabolic if 0Ads

K u . We also call it an Anti de Sitter flat point if it is

an umbilic point and 0Ads

K u .

3 SOME PROPERTIES OF THE TIMELIKE ANTI DE SITTER HEIGHT FUNCTION

In this section, we define a family of function on a spacelike hypersurface in Anti de Sitter 4-space, which are useful

for the study of singularities of TAdS Gauss image. Let 4:X U Ads be a spacelike hypersurface. We define a

family of functions.

4:H U AdS R

by

2

, ,H u v X u v

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We call H a timelike Anti de Sitter height function (or an AdS-height function) on M . We denote the Hessian

matrix of the AdS-height function 0 0

,v

h u H u v at 0

u by 0 0v

Hess h u .Then we have the following

proposition.

Proposition 3.1 Let ( )M X U be a spacelike hypersurface in 4Ads and ,H u v be an AdS-height function. Then

(1) , , 0 1,2,3i

HH u v u v i

u

If and only if v N u ;

(2) Let 0 0v N u , then

0 0det 0

vHess h u if and only if 0

AdsK u .

Proof (1) There exist real numbers 1 2 3

, , , , such that

1 2 31 2 3u u uv X N X X X

Since2

, 1X X , we get2

, 0iu

X X . It follows that ( , ) 0H u v if and only if 2

, 0X u v . Since

3

2 1

0 , ,iu ij i

ji

Hu v X u v g

u

And ijg is non-degenerate, we get 0 1,2,3

ii . It follows that v N . We also get

2, 1v v .

Therefore 1 .

(2) By definition, we get

0 0 0 0 02 2

, ,i j i jv u u u u

Hess h u X N u X u N u

By the AdS-Weingarten formula, we get

3 3

2 21 1

, ,i j ju u i u u i ij

X N h X X h g j h

.

Therefore we get

0 0

0

detdet

det det

vij

Ads

Hess h uhK

g g u

Then we complete the proof.

Corollary 3.2 Let 4:H U AdS R be an AdS-height function on spacelike hypersurface ( )M X U . N is the

TAdS-Gauss image and p M . Then the following conditions are equivalent:

(1) There exists 4v Ads such that p is a degenerate singular point of AdS-height functionv

h ;

(2) There exists 4v Ads such that p is a singular point of TAdS-Gauss image N ;

(3) 0Ads

K u .

Proof By definition, (2) and (3) are equivalent. By the assertion (2) of above proposition, (1) and (3) are also

equivalent.

Next, in order to study the TAdS-Gauss image of M , we should refer to the brief review on Legendrian singularity

theory in [3]. Now we apply the arguments in [2-3] to our situation. We can naturally interpret the TAdS-Gauss

image as a Legendrian map in the framework of Legendrian singularity theory.

Let 4:X U Ads be a spacelike hypersurface in 4Ads and N be the TAdS-Gauss image of ( )M X U . We define a

mapping

:L U by ,L u X u N u

where

4 4

2, , 0v w v w AdS Ads .

Since

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2 2

, , 0X u N u dX u N u

The mapping L is a Legendrian embedding. We denote 1 2 3 4 5, , , ,X u x x x x x and 1 2 3 4 5

, , , ,X u as

coordinate representations. We define a smooth mapping

* 4:U PT AdS

by

1 2 2 1 1 3 3 1 1 4 4 1 1 5 5 1( ,[ : : : ])u N u x x x x x x x x .

It is known that two fundamental notions of Legendrian singularity theory are Morse family and generating family.

We can get the following proposition on Morse family.

Proposition 3.3 The AdS-height function 4:H U AdS R is a Morse family.

Legendrian dualities between pseudo in general semi-Euclidean spaces have been studied in [3]. Next we will prove

H to be a generating family of ( )L U . Lengendrian dualities will be used as a fundamental tool. We get the

following proposition on generating family.

Proposition 3.4 For any spacelike hypersurfaces 4:X U AdS , the AdS-height function 4:H U AdS R of X is

a generating family of the Legendrian embedding L .

Proof We consider a coordinate neighbourhood

4

1 1 2 3 4 5 1, , , , 0W w w AdS

We also consider the contact morphism:

* 4

1 1: ( )W PT AdS W

Since AdS-height function H is a Morse Family, we can define a Legendrian immersion

* 4

* 1 1: ( )

HL H U W PT AdS W

by

2

( , ) ( ,[ :H

HL u

3 4 5

: : ])H H H

By Proposition 3.1, we have

4

*( ) { , ( ) }H u N u U AdS u U ,

where 4

*

1 2 3

( ) {( , ) , ( , ) ( , ) ( , ) 0H H H

H u U AdS H u u u uu u u

.

Since N u and1

2 2 2 2

2 3 4 51 , we get

2

2 1

2 1

( , )H

u N u x u x u

,

3

3 1

3 1

( , )H

u N u x u x u

,

4

4 1

4 1

( , )H

u N u x u x u

,

5

5 1

5 1

( , )H

u N u x u x u

,

where 1 2 3 4 5, , , ,X x x x x x and 1 2 3 4 5

, , , ,N . It follows that

1 2 2 1 1 3 3 1 1 4 4 1 1 5 5 1, ( ) ( ( ),[ : : : ])

HL u N u N u x x x x x x x x

( )u

Therefore we get ( ) ( )L u u . This indicates that H is a generating family of L .We can get the same

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relation as the above on the other local coordinates. Therefore we conclude that the TAdS-Gauss image N can be

regarded as a Legendrian map.

4 CONTACT WITH ADS-GREAT-HYPERBOLOIDS

In this section, we consider the geometric meaning of the singularities of the TAdS-Gauss image of spacelike

hypersurface ( )M X U in 4AdS . We should consider the contact between spacelike hypersurface and AdS-great-

hyperboloid. Montaldi [7] introduce some basic notions of contact geometry and he gave a characterization of the

notion of contact by using the terminology of singularity theory. For notions and basic results on the theory of

Legendrian singularities, refer to [5].

We define a function

4 4:H Ads Ads R

by

2

, ,H u v u x .

For any 4

0v AdS , we denote

0 0,

vu H u v and we have the AdS-great-hyperboloid

0

1 4

0(0) ( ,0)

vM AdS HP v . We write 4

0, 00 ( ,0)AH v AdS HP v . For any

0u U , we consider the timelike

vector 0 0

( )v N u .Then we get

40 0 0 0 0 0

, , 0v Ads

X u H X id u v H u N u .

We also get

0

0 0 0, 0

v

i i

X Hu u N u

u u

.

For 1,2,3i . This means that the AdS-great-hyperboloid 0,0AH v is tangent to M at

0( )P X u . In this case, we

call 0,0AH v the tangent AdS-great-hyperboloid of ( )M X U at

0( )P X u and we denote it by 0

,AH X u . Let

1 2,v v be timelike vectors. If

1v and

2v are linearly dependent, then 1

,0HP v and 2,0HP v are equal. Therefore,

1 2,0 ,0AH v AH v . We have the following simple lemma.

Lemma 4.1 Let 4:X U Ads be a spacelike hypersurface. For two points1 2,u u u ,

1 2( ) ( )N u N u if and only if

1 2, ,AH X u AH X u .

We now study the contact between M and tangent AdS-great-hyperboloid at p M as an application of Legendrian

singularity theory. We should introduce an equivalence relation among Lengendrian immersion germs. Let

*: , ,ni L p PT R p and ' ' ' * ': , ,ni L p PT R p be Legendrian immersion germs. Then we say that i and 'i are

Legendrian equivalent if there exists a contact diffeomorphism germ * * ': , ,n nH PT R P PT R p such that

H preserves fibres of land '( )H L L . A Legendrian germ into * nPT R at a point is said to be Legendrian stable if

for every map with the given germ there are a neighbourhood in the space of Legendrian immersion (in the Whitney

C topology) and a neighbourhood has, in the second neighborhood, a point at which its germ is Legendrian

equivalent to the original germ [1] The definition of -equivalent, P equivalent and equivalent in [5] will

be used here. We have the following theorem.

Theorem 4.2 Let 4: , , 1,2i i i i

X U u AdS X u i be spacelike hypersurface germs such that the

corresponding Legendrian embedding germs : , ,i

i iL U u z are Lengendrian stable. Then the following

conditions are equivalent.

(1) TAdS-Gauss image germs 1

N and 2

N are -equivalent;

(2) 1

H and 2

H are P equivalent ;

(3) 11,v

h and 22,v

h are equivalent;

(4) 1 1 1 1 2 2 2 2, , , , , ,X U AH X u v X U AH X u v ;

(5) 1 1,Q X u and 2 2

,Q X u are isomorphic as R-algebras, where

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2

,,

i

ui

u

i i

C

CQ X u

X u N u

.

Proof By the analogous arguments in [5], it is easy to get that these conditions are equivalent.

For a spacelike hypersurface germ

4

0 0: , ,X U u AdS X u ,

1

0 0,0 ,X AH N u u is called the tangent AdS-great-hyperboloidic indicatrix germ of X . In general we have

the following theorem:

Theorem 4.3 Let 4: , , 1,2i i i i

X U u AdS X u i be spacelike hypersurface germs such that their AdS-

parabolic sets have no interior an subspaces of U . If TAdS-Gauss image germs 1

N and 2

N are -equivalent, then

1 1 1 1 2 2 2 2, , , , , ,X U AH X u v X U AH X u v .

In this case, 1

1 1 1 1,0 ,X AH N u u and 1

2 2 2 2,0 ,X AH N u u are diffeomorphic as set germs.

Proof The AdS-parabolic set is the set of singular points of the TAdS-Gauss image. So the corresponding

Legendrian embedding iL satisfies the conditions of legendrian stability. If TAdS-Gauss image germs 1

N and 2

N

are

-equivalent, then 1L and 2L are Lengendrian equivalent, so that 1

H and 2

H are P equivalent. Therefore,

11vh and

22vh are equivalent. This condition is equivalent to the condition that

1 1 1 1 2 2 2 2, , , , , ,X U AH X u v X u AH X u v .

On the other hand, we have

1 1

0 0 0( ( ),0 , ) 0 ,

ii i ivX AH N u u h u

Since the equivalent preserves the zero level sets, we get that 1

1 1 1( ( ),0 , )

iX AH N u u and

1

2 2 2 2( ( ),0 , )X AH N u u are diffeomorphic as set germs.

Theorem 4.3 means the diffeomorphism type of the tangent AdS-great-hyperboloidic indicatrix germ is an invariant

of A-classification of the TAdS-Gauss image germ of X

5 GENERIC PROPERTIES OF SPACELIKE HYPERSURFACES

In this section, we consider generic properties of spacelike hypersurfaces in 4Ads . A kind of transversality theorem

will be used as a main tool. We consider the space of spacelike embeddings Embs 4( , )U AdS with Whitney C -

topology. We also consider the function 4 4:H AdS AdS R defined in §4. We claim that u

H is a submersion for

any 4u AdS , where ,u

H v H u v . For any X Emb 4( , )U AdS , we have 4AdsH H X id . We also have

the l jet extension 4

1: ,l lj H U AdS J U R defined by 1

( , )l l

vj H u v J h u . We consider the

trivialization , 3,1l lJ U R U R J . For any submanifolds 3,1lQ J , we denote~

{0}Q U Q . Then we

have the following proposition and a corollary of lemma 6 of Wassermann [6].

Proposition 5.1 Let Q be a submanifold of (3,1).lJ then the set

~4

1{ ( , ) }l

QT X Embs U Ads j H is transversality to Q

is a residual subset of Embs4( , )U AdS . If Q is a closed subset, then

QT is open.

By the classification of stable Legendrian singularities of 6n and Proposition 5.1, we have the following theorem.

Theorem 5.2 There exists an open dense subset o Embs4( , )U AdS such that for any X O , the germ of the

corresponding Legendrian embedding L at each point is Legendrian stable.

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ACKNOWLEDGEMENTS

The author was partially supported by preparatory studies of provincial Innovation project of MNU, No.SY201225.

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[4] Kong L L, Pei D H. Singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space [J]. Science in China Ser

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[6] Wassermann G. Stability of caustics [J]. Mathematische Annalen, 1975, 210: 43-50.

[7] Montaldi J A. On generic composites of maps [J]. Bulletin London Mathematical Society, 1991, 23: 81-85.

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AUTHORS

Jiajing Miao was born in Taonan, Jilin, China in 1982. She received the B.S and the M.S. degrees from Jilin

Normal University in 2004 and 2007, respectively.

Since 2009, she has been a lecturer in Mudanjiang Normal University. Miao is mainly interested in Geometry and

Lie algebra. She has published more than ten papers.

Email: jiajing0407@126.com