Differentiable Manifolds/Diffeomorphisms and related vector fields
We shall now define the notions of homeomorphisms and diffeomorphisms for mappings between manifolds.
The rank of the differential edit
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.
Related vector fields edit
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 :
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 .