# Differentiable Manifolds/Diffeomorphisms and related vector fields
## Diffeomorphisms

We shall now define the notions of homeomorphisms and diffeomorphisms for mappings between manifolds.
Definition 6.1:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where as usual $n\in \mathbb {N} _{0}\cup \{\infty \}$, and let $N$ be a manifold of class ${\mathcal {C}}^{k}$, where also $k\in \mathbb {N} _{0}\cup \{\infty \}$. A function $\psi :M\to N$ is called a homeomorphism iff
* it is bijective
* both itself and $\psi ^{-1}$ are continuous
Definition 6.2:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$, and let $N$ be a manifold of class ${\mathcal {C}}^{k}$, where also $k\in \mathbb {N} _{0}\cup \{\infty \}$. Let $j\in \mathbb {N} \cup \{\infty \}$. A function $\psi :M\to N$ is called a diffeomorphism of class ${\mathcal {C}}^{j}$ iff
* it is bijective
* both itself and $\psi ^{-1}$ are differentiable of class ${\mathcal {C}}^{j}$

## The rank of the differential

Definition 6.3:
Let $d,b\in \mathbb {N}$, $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, $N$ be a $b$-dimensional manifold of class ${\mathcal {C}}^{k}$, $U\subseteq M$ be open, $p\in U$ and $\psi :U\to N$ differentiable of class ${\mathcal {C}}^{m}$. The rank of $d\psi _{p}$ is defined as
    ${\text{rank }}d\psi _{p}:=\dim {\text{Im }}d\psi _{p}$.
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.
Theorem 6.4:
Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$ and $N$ be a $b$-dimensional manifold of class ${\mathcal {C}}^{k}$.

## Related vector fields

Definition 6.3:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$, let $N$ be a manifold of class ${\mathcal {C}}^{k}$, where also $k\in \mathbb {N} _{0}\cup \{\infty \}$, and let $\psi :M\to N$ be differentiable of class ${\mathcal {C}}^{j}$, where also $j\in \mathbb {N} \cup \{\infty \}$. We call $\mathbf {V} \in {\mathfrak {X}}(M)$ and $\mathbf {W} \in {\mathfrak {X}}(N)$ $\psi$-related iff
    $\forall p\in M:\mathbf {W} (\psi (p))=d\psi _{p}(\mathbf {V} (p))$
Theorem 6.4:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$, let $N$ be a manifold of class ${\mathcal {C}}^{k}$, where also $k\in \mathbb {N} _{0}\cup \{\infty \}$, let $\psi :M\to N$ be a diffeomorphism of class ${\mathcal {C}}^{j}$, where also $j\in \mathbb {N} \cup \{\infty \}$, and let $\mathbf {V} \in {\mathfrak {X}}(M)$. Then $\mathbf {W}$
    $\mathbf {W} (q)=d\psi _{\psi ^{-1}(q)}(\mathbf {V} (\psi ^{-1}(q)))$
is the unique vector field such that $\mathbf {V}$ and $\mathbf {W}$ are $\psi$-related.
Proof:
1. We show that $\mathbf {V}$ and $\mathbf {W}$ are $\psi$-related.
Let $p\in M$ be arbitrary. Then we have:
    $\mathbf {W} (\psi (p))=d\psi _{\psi ^{-1}(\psi (p))}(\mathbf {V} (\psi ^{-1}(\psi (p))))=d\psi _{p}(\mathbf {V} (p))$
2. We show that there are no other vector fields besides $\mathbf {W}$ which are $\psi$-related to $\mathbf {V}$.
Let $\mathbf {Z}$ be also contained in ${\mathfrak {X}}(N)$ such that $\mathbf {V}$ and $\mathbf {Z}$ are $\psi$-related. We show that $\mathbf {Z} =\mathbf {W}$, thereby excluding the possibility of a different to ${\mathcal {X}}$ $\psi$-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 $\mathbf {Z}$ was required to be $\psi$-related to $\mathbf {V}$:
    $\mathbf {Z} (q)=\mathbf {Z} (\psi (p))=d\psi _{p}(\mathbf {V} (p))=d\psi _{\psi ^{-1}(q)}(\mathbf {V} (\psi ^{-1}(q)))=\mathbf {W} (q)$
Theorem 6.5:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$, let $N$ be a manifold of class ${\mathcal {C}}^{k}$, where also $k\in \mathbb {N} _{0}\cup \{\infty \}$, let $\psi :M\to N$ be a diffeomorphism of class ${\mathcal {C}}^{j}$, where also $j\in \mathbb {N} \cup \{\infty \}$. If $\mathbf {V}$ is differentiable of class ${\mathcal {C}}^{m}$ where $m\in N$ and $m\leq j$, then the unique $\psi$-related vector field
    $\mathbf {W} (p)=d\psi _{\psi ^{-1}(q)}(\mathbf {V} (\psi ^{-1}(q)))$
is differentiable of class ${\mathcal {C}}^{m}$.
Proof:
Let $\vartheta \in {\mathcal {C}}^{m}(N)$. Inserting a few definitions from chapter 2, we obtain
    ${\begin{aligned}\mathbf {W} \vartheta (p)&=(d\psi _{\psi ^{-1}(p)}(\mathbf {V} (\psi ^{-1}(p))))(\vartheta )\\&=(\mathbf {V} (\psi ^{-1}(p))\circ \psi ^{*})(\vartheta )\\&=\mathbf {V} (\psi ^{-1}(p))(\vartheta \circ \psi )\\&=\mathbf {V} (\vartheta \circ \psi )(\psi ^{-1}(p))\\&=\left(\left(\psi ^{-1}\right)^{*}\mathbf {V} (\vartheta \circ \psi )\right)(p)\\\end{aligned}}$
, and therefore
    $\mathbf {W} \vartheta =\left(\psi ^{-1}\right)^{*}\mathbf {V} (\vartheta \circ \psi )$
Since $\psi$ is differentiable of class ${\mathcal {C}}^{j}$ and $j\geq m$, $\psi$ is also differentiable of class ${\mathcal {C}}^{m}$. Further, the function
    $\mathbf {V} (\vartheta \circ \psi )$
is differentiable of class ${\mathcal {C}}^{m}$, because $\mathbf {V}$ is differentiable of class ${\mathcal {C}}^{m}$. Due to lemma 2.17, it follows that also
    $\mathbf {W} \vartheta =\left(\psi ^{-1}\right)^{*}\mathbf {V} (\vartheta \circ \psi )$
is differentiable of class ${\mathcal {C}}^{m}$, and therefore, due to the definition of differentiability of vector fields, so is $\mathbf {W}$.$\Box$