Differentiable Manifolds/Diffeomorphisms and related vector fields

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Differentiable Manifolds
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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 .

Related vector fields[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
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