Differentiable Manifolds/Diffeomorphisms and related vector fields

Diffeomorphisms

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

The rank of the differential

The dimension of $\text{Im }d{\psi }_p$ is well-defined since $d{\psi }_p$ is a linear function, which is why it's image is a vector space; further, it is a vector subspace of $T_{\psi (p)}N$, which is a $b$-dimensional vector space, which is why it has finite dimension.

Related vector fields

Proof:

1. We show that $𝐕$ and $𝐖$ are $\psi$-related.

Let $p\in M$ be arbitrary. Then we have:

$𝐖(\psi (p))=d{\psi }_{{\psi }^{-1}(\psi (p))}(𝐕({\psi }^{-1}(\psi (p))))=d{\psi }_p(𝐕(p))$

2. We show that there are no other vector fields besides $𝐖$ which are $\psi$-related to $𝐕$.

Let $𝐙$ be also contained in $𝔛(N)$ such that $𝐕$ and $𝐙$ are $\psi$-related. We show that $𝐙=𝐖$, thereby excluding the possibility of a different to $𝒳$ $\psi$-related vector field.

Indeed, for every $q\in N$ we have:

Due to the bijectivity of $\psi$, there exists a unique $p\in M$ such that $\psi (p)=q$, and we have $p={\psi }^{-1}(q)$. Therefore, and since $𝐙$ was required to be $\psi$-related to $𝐕$:

$𝐙(q)=𝐙(\psi (p))=d{\psi }_p(𝐕(p))=d{\psi }_{{\psi }^{-1}(q)}(𝐕({\psi }^{-1}(q)))=𝐖(q)$

Proof:

Let $\vartheta \in {𝒞}^m(N)$. Inserting a few definitions from chapter 2, we obtain

$\begin{array}{cc}𝐖\vartheta (p)& =(d{\psi }_{{\psi }^{-1}(p)}(𝐕({\psi }^{-1}(p))))(\vartheta )\\ & =(𝐕({\psi }^{-1}(p))\circ {\psi }^{*})(\vartheta )\\ & =𝐕({\psi }^{-1}(p))(\vartheta \circ \psi )\\ & =𝐕(\vartheta \circ \psi )({\psi }^{-1}(p))\\ & =({\left({\psi }^{-1}\right)}^{*}𝐕(\vartheta \circ \psi ))(p)\\ \end{array}$

, and therefore

$𝐖\vartheta ={\left({\psi }^{-1}\right)}^{*}𝐕(\vartheta \circ \psi )$

Since $\psi$ is differentiable of class ${𝒞}^j$ and $j\ge m$, $\psi$ is also differentiable of class ${𝒞}^m$. Further, the function

$𝐕(\vartheta \circ \psi )$

is differentiable of class ${𝒞}^m$, because $𝐕$ is differentiable of class ${𝒞}^m$. Due to lemma 2.17, it follows that also

$𝐖\vartheta ={\left({\psi }^{-1}\right)}^{*}𝐕(\vartheta \circ \psi )$

is differentiable of class ${𝒞}^m$, and therefore, due to the definition of differentiability of vector fields, so is $𝐖$.$◻$