Differentiable Manifolds/Diffeomorphisms and related vector fields

Differentiable Manifolds
 ← Vector fields, covector fields, the tensor algebra and tensor fields Diffeomorphisms and related vector fields Lie algebras and the vector field Lie bracket → 

Diffeomorphisms

edit

We shall now define the notions of homeomorphisms and diffeomorphisms for mappings between manifolds.

Definition 6.1:

Let   be a manifold of class  , where as usual  , and let   be a manifold of class  , where also  . A function   is called a homeomorphism iff

  • it is bijective
  • both itself and   are continuous

Definition 6.2:

Let   be a manifold of class  , where  , and let   be a manifold of class  , where also  . Let  . A function   is called a diffeomorphism of class   iff

  • it is bijective
  • both itself and   are differentiable of class  

The rank of the differential

edit

Definition 6.3:

Let  ,   be a  -dimensional manifold of class  ,   be a  -dimensional manifold of class  ,   be open,   and   differentiable of class  . The rank of   is defined as

 .

The dimension of   is well-defined since   is a linear function, which is why it's image is a vector space; further, it is a vector subspace of  , which is a  -dimensional vector space, which is why it has finite dimension.

Theorem 6.4:

Let   be a  -dimensional manifold of class   and   be a  -dimensional manifold of class  .

edit

Definition 6.3:

Let   be a manifold of class  , where  , let   be a manifold of class  , where also  , and let   be differentiable of class  , where also  . We call   and    -related iff

 

Theorem 6.4:

Let   be a manifold of class  , where  , let   be a manifold of class  , where also  , let   be a diffeomorphism of class  , where also  , and let  . Then  

 

is the unique vector field such that   and   are  -related.

Proof:

1. We show that   and   are  -related.

Let   be arbitrary. Then we have:

 

2. We show that there are no other vector fields besides   which are  -related to  .

Let   be also contained in   such that   and   are  -related. We show that  , thereby excluding the possibility of a different to    -related vector field.

Indeed, for every   we have:

Due to the bijectivity of  , there exists a unique   such that  , and we have  . Therefore, and since   was required to be  -related to  :

 

Theorem 6.5:

Let   be a manifold of class  , where  , let   be a manifold of class  , where also  , let   be a diffeomorphism of class  , where also  . If   is differentiable of class   where   and  , then the unique  -related vector field

 

is differentiable of class  .

Proof:

Let  . Inserting a few definitions from chapter 2, we obtain

 

, and therefore

 

Since   is differentiable of class   and  ,   is also differentiable of class  . Further, the function

 

is differentiable of class  , because   is differentiable of class  . Due to lemma 2.17, it follows that also

 

is differentiable of class  , and therefore, due to the definition of differentiability of vector fields, so is  . 

Differentiable Manifolds
 ← Vector fields, covector fields, the tensor algebra and tensor fields Diffeomorphisms and related vector fields Lie algebras and the vector field Lie bracket →