Differentiable Manifolds/Vector fields, covector fields, the tensor algebra and tensor fields

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Differentiable Manifolds
 ← Submanifolds Vector fields, covector fields, the tensor algebra and tensor fields Diffeomorphisms and related vector fields → 

In this section, the concepts of vector fields, covector fields and tensor fields shall be presented. We will also define what it means that one of those (vector field, covector field, tensor field) is differentiable. Then we will show how suitable restrictions of all these things can be written as sums of the bases of the respective spaces induced by a chart, and we will show a simultaneously sufficient and necessary condition of differentiability based on this sum expression.

Vector fields[edit]

Definitions 5.1:

Let be a manifold, let be open, and let

be a function from to the tangent bundle of , such that:

Then we call a vector field on .

If and are two vector fields and , we define

The set of all vector fields on we denote by .

Definition 5.2:

Let be a manifold of class , let be open, let be a vector field on and let . We call differentiable of class iff for all in the function

is contained in .

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

Lemma 5.4:

Let be a -dimensional manifold of class and be contained in its atlas. Then the vector fields

are differentiable of class .

Proof:

Let . Then we have:

Let now be another chart in the atlas of . Then the function

is smooth, as the composition of smooth functions.

If is a manifold of class , we even have that since is contained in for all , if a chart around is given by , then for all

, where

are functions from to . This follows from chapter 2, where we remarked based on two theorems of the section, that

is a basis of for .

Theorem 5.5:

Let be a vector field on the open subset of the -dimensional manifold , and let be contained in the atlas of . Then the vector field is differentiable of class iff all the , as defined by eq. above, are contained in .

Proof:

1.) We prove that if all the defined by are contained in , that then is differentiable of class .

This is because if is contained in , then due to lemma 5.4 and theorem 2.24 all the summands of the function

are differentiable of class . Due to theorem 2.23 and induction, we have that the function itself is differentiable of class . Due to , the function is identical to .

2.) We prove that if is differentiable of class , then so are the defined by .

Due to lemma 2.3, if we write , the functions , are contained in .

By definition of the differentiability of class of , we have that the functions

are contained in . But due to and lemma 2.4, we have for all :

Hence:

, where since the two functions are equal and one of them is differentiable of class , both of them are differentiable of class .

Covector fields[edit]

Definition 5.6:

Let be a manifold, let be open, and let

be a function from to the cotangent bundle of , such that:

Then we call a covector field on .

Definition 5.7:

Let be a manifold, let be open and let be a covector field. Then we call differentiable of class iff for all vector fields which are differentiable of class , we have that the function

is contained in .

Lemma 5.9:

Let be a manifold of class and be contained in its atlas. Then the covector fields

are differentiable of class .

Proof:

Let be differentiable of class , and let . Due to lemma 2.3, the function is differentiable of class . Since is differentiable of class , it follows that

is differentiable of class (the latest equation follows from the definition of ).

If is a manifold of class , we even have that since is contained in for all , if a chart around is given by , then for all

, where

are functions from to . This follows from chapter 2, where we remarked based on two theorems of the chapter, that

is a basis of for .

Theorem 5.10:

Let be a covector field on the -dimensional manifold . Then is differentiable of class iff all the , , as defined in equation , are contained in .

Proof:

1.) We show that the differentiability of class of the defined by implies the differentiability of .

Let be a vector field on which is differentiable of class . Due to lemma 5.9 and theorem 2.24, all the summands of the function

are contained in . Therefore, due to theorem 2.23 and induction, also the function itself is contained in . But due to , the function is equal to .

2.) We show that if is differentiable, then so are the , defined by .

Due to lemma 5.4, we have that for , the vector field is differentiable of . Hence, due to the differentiability of , the function

is contained in . But due to , we have

Hence:

and hence .

The tensor algebra[edit]

Definition 5.9:

By an algebra, one often means a vector space such that there is a function which is bilinear, i. e. satisfies for all and :

and

Definition 5.11:

Let be a vector space, and its dual space.

Tensor fields[edit]

Definition 5.?:

Differentiable Manifolds
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