Global Theory of Smooth Functions of Manifolds · infinitesimal G-structures and integrable -structures. For conceptual simplicity, all manifolds, functions, bundles, vector fields,

International Journal of Interdisciplinary Research in Science Society and Culture(IJIRSSC) Vol: 2, Issue:1, (June Issue), 2016 ISSN: (P) 2395-4345, (O) 2455-2909 © IJIRSSC

www.ijirssc.in Page 408

Global Theory of Smooth Functions of

Manifolds

Mohammad Raquibul Hossain1,*

, Md. Raknuzzaman2 , Md. Mijanoor

Rahman3, Md. Jamal Hossain

4

1,4Department of Applied Mathematics, Noakhali Science & Technology University,

Noakhali-3814, Bangladesh. E-mail:[email protected] and [email protected] 2Department of Arts and Sciences, Ahsanullah University of Science and Technology, Dhaka,

Bangladesh. Email: [email protected]

3Department of Mathematics, Mawlana Bhasani Science & Technology University, Santos,

Tangail-1902, Bangladesh. Email: [email protected]

ABSTRACT : Our present goal is to extend the theory of smooth functions, developed on open

subsets of in the local theory of smooth functions to arbitrary differentiable Manifolds, in

this case geometric topology becomes an essential feature.

Keywords: Algebraic Topology, Differential Geometry, Manifold, Topology.

I. Introduction: Fundamental to the global theory of differentiable manifolds is the concept of a vector

bundle. The easiest nontrivial example is the tangent bundle to the -sphere which we

introduce from a purely topological point. The subtleties involved in this bundle are

illustrated in the discussion of the vector field problem. As the global theory is developed, the

tangent bundle, the cotangent bundle, various tensor bundles, and the associated (principal)

frame bundles will play increasingly important roles [3], as will the related notions of

infinitesimal G-structures and integrable -structures. For conceptual simplicity, all

manifolds, functions, bundles, vector fields, Lie groups, homogeneous spaces, etc., will be

smooth of class . We study Smooth manifolds and mapping, diffeomorphic structures, the

tangent bundle, cocycles and geometric structures, global construction of smooth functions,

manifolds with boundary and finally submanifolds.

II. Objectives of the Study: Manifold is a very important topic in the study of geometry, physics and quantum field

theory. There are many types of manifolds which are extensively used in those study areas.

Here in this paper we use -manifolds which are known as smooth manifolds and by using

smooth manifolds we emphasise global structure of functions which is the important topics

for study geometry.



III. Results and Discussion: 1. Smooth Manifolds and Mapping

Let be a topological manifold of dimension . The locally Euclidean property allows us to

choose local coordinates in any small region of

Definition 1.01 A coordinate chart [2] on is a pair , where is an open subset

and is a homeomorphism onto an open subset .

We usually write , viewing this as the coordinate -tuple of the

point

Definition 1.02 Two coordinate charts and on are said to be related if

either or

is diffeomorphism.

Definition 1.03 A atlas on is a collection of coordinate charts

such that

is related to ,

Two atlas and on are equivalent if is also a atlas on .

Example 1.04 The manifold has a canonical smooth structure [1], namely the set of

all pairs where is open and is a diffeomorphism onto an open set

.

Example 1.05 If is a smooth -manifold and a smooth -manifold with respective

smooth structures and , then is canonically a

smooth manifold. Indeed

is a atlas, determining uniquely a maximal one, called the Cartesian product of the two

smooth structures.

Definition 1.06 Let be a smooth -manifold with smooth atlas . Set

the function

These local diffeomorphisms in satisfying the cocycle conditions



It should be noted that the properties and follows from the property .

Definition 1.07 Let be a smooth -manifold then . On the

disjoint union

define the relation

such that , and .

By properties and of the definition 1.06, this is an equivalence relation, so we

form the topological quotient space . We will show that this space is homeomorphic to

and exhibit a natural smooth structure on it.

Let denote the equivalence class . Define

By setting if . If also then , so is

well defined. It is continuous. The map from to that takes to respects to

equivalence relation, hence passes to a continuous map

.

It is easy to see that and are mutually inverse, so and are canonically

homeomorphic. Each imbeds canonically in as an open subset and

defines a coordinate chart on . These charts are -related via the cocycle

, so is canonically identified with as a smooth manifold via the mutually

diffeomorphisms and .

Definition 1.08 A function is said to be smooth if, for each , there exist a

chart such that and

is smooth. The set of all smooth, real valued functions on will be denoted .

Lemma 1.09 The function is smooth if and only if

is smooth, .



Proof. Condition implies that is smooth. For the converse, suppose that is smooth and let

where . By definition 1.08, choose such that and

is smooth. Then

is given by the decomposition

as a composition of smooth maps, this is smooth. That is,

is smooth on some neighborhood of the point . But is arbitrary, so is

smooth on all of □

Definition 1.10 Let and be manifolds with respective smooth structures and . A

mapping is said to be smooth if, for each , there are and

such that , , and

is smooth.

Definition 1.11 A derivative of is a - linear map

such that

. This operator is called a tangent vector to at and the vector space

of all derivative of is called the tangent space [11] to at .

Definition 1.12 If is a smooth map between manifolds and if , the

differential

is the linear map defined by

,

for all and all .



Lemma 1.13 If and If are smooth map between manifolds and ,

then .

Proof. Consider

Since and are arbitrary. □

Corollary 1.14 If is a smooth manifold of dimension , then is a real vector space

of dimension , .

Proof. Let be a coordinate patch on with . Then so we have

is an - linear isomorphism. Since is open, we know that

□

2. Diffeomorphic Structures

Diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible

function that maps one differentiable manifold to another, such that both the function and its

inverse are smooth functions.

This section is really an extended remark on some very deep theorems. For this purpose, let

be a differentiable manifold with smooth structure [10] . Let be any homeomorphism. Set

.

Proposition 2.01 The set is a structure on having the same structure cocycle as .

Proof. We know

,

and this map carries the set

onto the set



Finally, the maximality of the atlas follows from that of and the fact that

. □

Definition 2.02 Two structures [6] and , defined on some topological manifold ,

are said to be diffeomorphic structures , for some homeomorphism .

Example 2.03 Let denote the number of diffeomorphism classes of differentiable

structures on . It was long known that for . The value of remains

mystery. The following table for was computed by Kervaire and Milnor.

5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1 28 2 8 6 992 1 3 2 16256 2 16 16

3. The Tangent Bundle

Let be a -maifold with structure . Consider the set

a disjoint union with, as set no topological structure. For each , define

.

Then the individual linear maps , , unite a define a set map

More precisely, if denotes a tangent vector to at

and this defines a bijection of onto an open subset of . If , consider

By the chain rule

a diffeomorphism between the open subsets of .

We topologize the set . If

is an open set, then is to be an open subset of .

Definition 3.01 The system is said called the tangent bundle [5] of . The total

space , the base space is , and is called the bundle projection.



Definition 3.02 A vector field on is a smooth map such that

. The set of all vector fields on is denoted by .

Remark: Let be a vector field on , a coordinate chart on . Then

where is smooth, .

Definition 3.03 Let be a smooth -manifold, a smooth - manifold, and

a smooth map. This will be called an -plane bundle over (or a vector bundle

over of fiber dimension ) if the following properties hold

For each , has the structure of real, -dimensional vector

space.

There is an open cover of , together with commutative diagrams

such that is diffeomorphism,

For each and maps the vector space isomorphically onto

the vector space .

As with tangent bundles, we call is the total space, is the base space and the bundle

projection. We also call each a trivializing a neighborhood for the bundle and a

locally trivializing cover (of ) for .

Definition 3.04 An -plane bundle is trivial if it is a isomorphic to the product bundle

.

Definition 3.05 A section of the -tuple is trivial is a smooth map such

that . The set of all such sections is denoted by .

Definition 3.06 The manifold parallelizable if there are fields

such that is a basis of , .

Example 3.07 We give 1-plane bundle (a line bundle) over a circle, known as the Möbius

bundle. On , define the equivalence relation . Remark

that is a linear automorphism of . The projection passes to a well



defined map . It should be clear, intuitively that this a vector

bundle over of fiber dimension one.

4. Cocycles and Geometric Structures

Let be an -plane bundle and let be a locally trivializing open cover of ,

the trivializations being . If , consider

.

This composition must have the form

,

where .

Lemma 4.01 The map is smooth.

Proof. Let denote the column vector with 0’s in all places except the , where the entry

1. Then defines a smooth map

,

. But and the second entry is just the column of

. Since is arbitrary, , we see that the entries of are the smooth

functions of . □

Definition 4.02 The smooth maps have the ‘cocycle’ property

.

. As usual, this property implies the following also, for all

appropriate choice of and indices .

Definition 4.03 A - cocycle property [4] on is a family such that

is an open cover of and is a smooth map, , all

subject to the cocycle condition property of definition 4.02. if the cocycle arises as

above from an -plane bundle , it is said to be a structure cocycle of .



Definition 4.04 Two -cocycles

and

On the same manifold equivalent if they are contained in a common -cocycle on .

The equivalence class of will be denoted by .

Corollary 4.05 Equivalence -cocycles is an equivalence relation.

Proof. If and , let be a cocycle containing both and , be cocycle containing

both and . Then and both contain . We know the theorem “If two -cocycles on

same manifold contain a common -cocycle, then they are contained in a common

-cocycle” from this theorem guarantees that they are contained in a common cocycle .

Then and , so . □

Theorem 4.06 If is a -cocycle on , the isomorphism class depends only one equivalence class . This defines a canonical bijective

correspondence

where denote the set of isomorphism classes of -plane bundles on and

denote the equivalence classes of -cocycles.

Proof. In this case we have to identify to . The tangent bundle

, any smooth atlas , with associated structure cocycle for ,

provides a structure cocycle . There are, of course structure cocycles for

that are not obtained in this way, but these special cocycles tie together the bundle

structure of and the smooth structure of . □

Definition 4.07 A structure cocycle for , associated to smooth atlas on

, will be called a Jacobian cocycle. If admits a Jacobian cocycle such that ,

, then is said to be integrably parallelizable.

Definition 4.08 Let be a subgroup and let be an -plane bundle. We

say that the structure group of can be reduced to if there is a -cocycle

representing the isomorphism class of such that . Such a cocycle will be

called a -cocycle for .

5. Global Constructions of Smooth functions

Proposition 5.01 Let be an -manifold, let be an open subset and a

compact subset. Then there is a smooth function such that and supp( ) .



Proof. For each , choose a coordinate neighborhood about , , and an

open, -dimensional interval with , centered at . Since is compact, finitely

many of the cover . □

Lemma 5.02 Let be a smooth manifold and let . Then the natural map

that carries is surjective.

Proof. Given , find with supp( ) dom( ) and on some

compact neighborhood of in dom( ). Then extends by 0 to the smooth function on

and . □

Proposition 5.03 If is an open cover of , there is a partition of unity

subordinate to .

Proof. First remark that is a refinement of unity subordinate to induces a

smooth partition of unity subordinate of . Indeed, let be a map such that

, . If is a partition of unity subordinate to , define ,

. If , we understand that . It is clear that is a partition of

unity subordinate to .

By passing to a suitable locally finite refinement of and applying the above paragraph, we

lose no generality in assuming that

is locally finite.

each is the domain of a coordinate chart ;

there are open -dimensional intervals with compact and such

that covers .

By Proposition 5.01, define in such a way that and supp( )

. Since the cover is locally finite, we can define

,

remarking that on . We obtain the smooth partition of unity by setting . □

Proposition 5.04 If is a smooth manifold and , then

is smooth if and only if , an open neighborhood of , and a smooth map

such that .

Proof. This property clearly follows from our definition of smoothness. We must recover our

definition from this property. Let . Then there is a must partition of unity



on , subordinate to the open cover of the manifold . Since each in -

valued, makes sense and can be interpreted as a smooth map of into . Then define

,

a smooth map of into . Evidently

,

, so is the required smooth extension of to the neighborhood of . □

Definition 5.05 A function from the subset of smooth manifold [7] into

the subset of a smooth manifold is said to be smooth if, for each , there is a

open neighborhood of and smooth map such that . Such a map is diffeomorphism of onto if it is bijective and both and

are smooth.

Theorem 5.06 Let and be manifolds of the same dimension . If is open, if

and if is a diffeomorphism, then is open in .

Proof. Let and . Since is smooth, there is an open

neighborhood of in and a smooth extension of . Since

is continuous, is an open neighborhood of in and

.

Since is smooth in the usual sense, the chain rules gives

,

so is a linear isomorphism. By the inverse function theorem,

there is a open neighborhood of which is carried by diffeomorphically onto

an open subset . But is an arbitrary point of and , so

is an open subset of . □

6. Manifolds with Boundary

Since Euclidean half space is a subset of smooth manifold , definition 5.05

allows us to talk about a smooth maps and diffeomorphism [8] between open subsets of .

Thus, if is a topological manifold with boundary, it makes sense to talk about two -

charts on being -related, so we can define a differentiable structure on to be a

maximal -atlas of -related charts. As in the topological case, we define

int( )

The pair is a (smooth) -manifold with boundary . Of course, all smooth -

manifolds without boundary are special cases, as is itself. The notation of smooth maps

between manifolds with boundary is defined exactly as in the boundaryless case.



Definition 6.01 If and are smooth maps between manifolds with

boundary, then is smooth and for each .

Lemma 6.02 Let and let

be defined by . Then is a surjection.

Proof. Let be an open neighborhood of . If is smooth, there is a

neighborhood of in and a smooth extension of . Then

and . □

Lemma 6.03 For , define

then by setting . Then we have have to show that is bijective.

Proof. We prove that is one to one. If , then

. Since is surjective, it follows that

, so .

We have to prove that is onto. Let . As an infinitesimal curve, this

vector is represented by . As an operator on germs, . Either points

into (we intend this to include the case that is tangent to ) or points out of , in

this case – points into .

If points into , then , . Define by

It is elementary that and that .

If points out of , then . Define by

Again, and . □

7. Submanifolds

Let be an -manifold, possibly with the boundary. A subset is a properly

imbedded submanifold [1] of dimension if and only if, , there is an coordinate

chart about in in which , where is the (image

of the) standard inclusion.

Remark that, in the above definition can be viewed as an coordinate

chart on and that the collection of all such charts makes a smooth -manifold with

boundary . Thus if , then also.



Theorem 7.01 Let be a smooth map between the manifolds of respective

dimensions and . Assume that and that . If is a regular value

simultaneously for for , then is a properly imbedded submanifold of

dimension .

Proof. Let and the find a suitable coordinate chart about in . There are two

cases.

Case 1. Suppose int( ). Choose a coordinate neighborhood about such that

int( ). Then is a regular value of so it implies that is a smooth

submanifold of of a dimension .

Case 2. Suppose and let be an charts about in . Assume

that , where is a coordinate chart about in which . Let

be denoted by with component functions relative

to the coordinate of . Since is a regular point for , can be chosen so small that the

matrix

has constant rank on . By permutation of coordinates , it can be

assumed that the last block

is non singular on . Choosing even smaller, if necessary the corresponding

block in the matrix

is also nonsingular. We then resort to the trick of recoordinatizing near by setting , , and . The inverse function theorem shows, by

the above remarks, that this will define an chart on a small enough neighborhood (again

called ) of . But, relative to these coordinates,

Then is the set of points with coordinates . That is

. □



Example 7.02 Let be given by

Then is a singular value both for and for . The hemisphere is the

intersection and is an - manifold with boundary [4] .

Lemma 7.03 If and is smooth, then the set of points in that are

simultaneous regular values for and in dense in .

Proof. Clearly, if is a regular point for , it is also the regular point for . Thus

is a regular value both of and precisely when it is a regular value both of

and of . Use countable coordinate coverings of int( ), of , and

of . For each , consider the countable family of smooth maps

Obtained by restrictions. So we have is a common regular value of all these maps.

Doing this for each , we complete proof. □

IV. Conclusion:

Differentiable manifolds form one of the key bases for developing geometry specially

in the study of differential geometry. Here in the paper we discuss seven subsections which

are the key bases for study to the other supplementary structures. The last section refers that a

transversal mapping along a submanifold plays an important role in problems of stability and

in the study of typical singularities of mappings in the study of geometry. Finally the

reference frames of -manifolds fit together globally which is of significant essence and

this feature contributes in solving problems of geometric topology related to global theory

case.

References:

[1] Brickell, F. and Clark. Differential Manifolds, Van Nostrand Reinhold Company,

London., R.S.1970

[2] Hilton, P.J. and Wylie, S., “Homology Theory”. Cambridge University Press, 1962.

[3] Kobayashi, S. and Nomizu, K., “Foundations of Differential Geometry”. Vol. I,

Interscience, 1969.

[4] Kobayashi, S. and Nomizu, K., “Foundations of Differential Geometry”. Vol. II,

Interscience, 1969.

[5] Wells, R. O., “Differential Analysis on Complex Manifolds”. Springer Verlag, 1979.

[6] Spanier, E. H., “Algebraic Topology”. P. 398. McGraw-Hill, 1966.

[7] Steenrod, N., “The Topology of Fibre Bundles”. Princeton University Press, 1970.

[8] Chevally, C., “Theory of Lie Groups”. Princeton University Press, 1946.

[9] Milnor, J. W., and Stasheff, J. D., “Characteristic Classes”. Princeton University



Press, 1974.

[10] Steenrod, N., “The Topology of Fibre Bundles”. Princeton University Press, 1970.

[11] Massey, W.S., “Singular Homology Theory”. Springer Verlag, 1980.

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* Corresponding Author: Mohammad Raquibul Hossain, Department of Applied

Mathematics, Noakhali Science & Technology University, Noakhali-3814, Bangladesh. E-mail:[email protected]

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Global Theory of Smooth Functions of Manifolds · infinitesimal G-structures and integrable -structures. For conceptual simplicity, all manifolds, functions, bundles, vector fields,