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

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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

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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

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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

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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

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Definition 5.?:

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