# 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

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

Definition 6.1:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where as usual ${\displaystyle n\in \mathbb {N} _{0}\cup \{\infty \}}$ , and let ${\displaystyle N}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ , where also ${\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}$ . A function ${\displaystyle \psi :M\to N}$  is called a homeomorphism iff

• it is bijective
• both itself and ${\displaystyle \psi ^{-1}}$  are continuous

Definition 6.2:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} _{0}\cup \{\infty \}}$ , and let ${\displaystyle N}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ , where also ${\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}$ . Let ${\displaystyle j\in \mathbb {N} \cup \{\infty \}}$ . A function ${\displaystyle \psi :M\to N}$  is called a diffeomorphism of class ${\displaystyle {\mathcal {C}}^{j}}$  iff

• it is bijective
• both itself and ${\displaystyle \psi ^{-1}}$  are differentiable of class ${\displaystyle {\mathcal {C}}^{j}}$

## The rank of the differential

Definition 6.3:

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

${\displaystyle {\text{rank }}d\psi _{p}:=\dim {\text{Im }}d\psi _{p}}$ .

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

Theorem 6.4:

Let ${\displaystyle M}$  be a ${\displaystyle d}$ -dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$  and ${\displaystyle N}$  be a ${\displaystyle b}$ -dimensional manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ .

Definition 6.3:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} _{0}\cup \{\infty \}}$ , let ${\displaystyle N}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ , where also ${\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}$ , and let ${\displaystyle \psi :M\to N}$  be differentiable of class ${\displaystyle {\mathcal {C}}^{j}}$ , where also ${\displaystyle j\in \mathbb {N} \cup \{\infty \}}$ . We call ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$  and ${\displaystyle \mathbf {W} \in {\mathfrak {X}}(N)}$  ${\displaystyle \psi }$ -related iff

${\displaystyle \forall p\in M:\mathbf {W} (\psi (p))=d\psi _{p}(\mathbf {V} (p))}$

Theorem 6.4:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} _{0}\cup \{\infty \}}$ , let ${\displaystyle N}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ , where also ${\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}$ , let ${\displaystyle \psi :M\to N}$  be a diffeomorphism of class ${\displaystyle {\mathcal {C}}^{j}}$ , where also ${\displaystyle j\in \mathbb {N} \cup \{\infty \}}$ , and let ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$ . Then ${\displaystyle \mathbf {W} }$

${\displaystyle \mathbf {W} (q)=d\psi _{\psi ^{-1}(q)}(\mathbf {V} (\psi ^{-1}(q)))}$

is the unique vector field such that ${\displaystyle \mathbf {V} }$  and ${\displaystyle \mathbf {W} }$  are ${\displaystyle \psi }$ -related.

Proof:

1. We show that ${\displaystyle \mathbf {V} }$  and ${\displaystyle \mathbf {W} }$  are ${\displaystyle \psi }$ -related.

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

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

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

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

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

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

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

Theorem 6.5:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} _{0}\cup \{\infty \}}$ , let ${\displaystyle N}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$ , where also ${\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}$ , let ${\displaystyle \psi :M\to N}$  be a diffeomorphism of class ${\displaystyle {\mathcal {C}}^{j}}$ , where also ${\displaystyle j\in \mathbb {N} \cup \{\infty \}}$ . If ${\displaystyle \mathbf {V} }$  is differentiable of class ${\displaystyle {\mathcal {C}}^{m}}$  where ${\displaystyle m\in N}$  and ${\displaystyle m\leq j}$ , then the unique ${\displaystyle \psi }$ -related vector field

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

is differentiable of class ${\displaystyle {\mathcal {C}}^{m}}$ .

Proof:

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

{\displaystyle {\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

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

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

${\displaystyle \mathbf {V} (\vartheta \circ \psi )}$

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

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

is differentiable of class ${\displaystyle {\mathcal {C}}^{m}}$ , and therefore, due to the definition of differentiability of vector fields, so is ${\displaystyle \mathbf {W} }$ .${\displaystyle \Box }$

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