# Differentiable Manifolds/Group actions and flows

 Differentiable Manifolds ← Integral curves and Lie derivatives Group actions and flows Product manifolds and Lie groups →

## Definitions of group actions and flows

Definition 9.1:

Let ${\displaystyle X}$  be a set, ${\displaystyle G}$  be a group, and ${\displaystyle e}$  the identity of ${\displaystyle G}$ . A group action is a function ${\displaystyle *:G\times X\to X}$  such that for all ${\displaystyle g,h\in G}$  and all ${\displaystyle x\in X}$ :

• ${\displaystyle (gh)*x=g*(h*x)}$
• ${\displaystyle e*x=x}$

Definition 9.2:

Let ${\displaystyle X}$  be a set. A flow on ${\displaystyle X}$  is a group action whose group is ${\displaystyle (\mathbb {R} ,+)}$ .

## The flow of a vector field

Definition 9.3:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} \cup \{\infty \}}$  (${\displaystyle n}$  must be ${\displaystyle \geq 1}$  here), and let ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$ . Due to theorem 8.2, for each ${\displaystyle p\in M}$  exists a maximal open interval ${\displaystyle I_{p}}$  such that ${\displaystyle 0\in I_{p}}$  and such that there is a unique curve ${\displaystyle \gamma _{p}:I_{p}\to M}$  such that ${\displaystyle \gamma _{p}(0)=p}$  and ${\displaystyle \gamma _{p}}$  is an integral curve of ${\displaystyle \mathbf {V} }$ . Then the flow of ${\displaystyle \mathbf {V} }$  is defined as the function

${\displaystyle \Phi _{\mathbf {V} }:\{(x,p)|p\in M,x\in I_{p}\}\to M,\Phi _{\mathbf {V} }(x,q):=\gamma _{q}(x)}$

Further, for all ${\displaystyle x\in I_{p}}$ , we define the function

${\displaystyle \Phi _{\mathbf {V} ,x}:\{p\in M|x\in I_{p}\}\to M,\Phi _{\mathbf {V} ,x}(q):=\Phi _{\mathbf {V} }(x,q)}$

Theorem 9.4: Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle n\in \mathbb {N} \cup \{\infty \}}$  (${\displaystyle n}$  must be ${\displaystyle \geq 1}$ ), let ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$  and let ${\displaystyle \Phi _{\mathbf {V} }}$  be the flow of ${\displaystyle \mathbf {V} }$ . If for each ${\displaystyle p\in M}$  the interval ${\displaystyle I_{p}}$  such that there is a unique curve ${\displaystyle \gamma _{p}:I_{p}\to M}$  such that ${\displaystyle \gamma _{p}(0)=p}$  and ${\displaystyle \gamma _{p}}$  is an integral curve of ${\displaystyle \mathbf {V} }$  is equal to ${\displaystyle \mathbb {R} }$ , then the flow of ${\displaystyle \mathbf {V} }$  is a flow.

Proof:

Let ${\displaystyle p\in M}$  be arbitrary.

1.

If we choose ${\displaystyle (O,\phi )}$  in the atlas of ${\displaystyle M}$  such that ${\displaystyle \gamma _{p}(y)\in O}$  and further define

${\displaystyle \rho _{p}:\mathbb {R} \to M,\rho _{p}(x):=\gamma _{p}(y+x)}$

, then using the fact that ${\displaystyle \gamma _{p}}$  is an integral curve of ${\displaystyle \mathbf {V} }$ , we obtain for all ${\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}$ , that

${\displaystyle (\varphi \circ \rho _{p})'(x)=(\varphi \circ \gamma _{p})'(x+y)=(\gamma _{p})_{x+y}'(\varphi )=\mathbf {V} (\gamma _{p}(x+y))(\varphi )=\mathbf {V} (\rho _{p}(x))(\varphi )}$

Hence, since ${\displaystyle \rho _{p}}$  and ${\displaystyle \gamma _{\gamma _{p}(y)}}$  are both integral curves and furthermore

${\displaystyle \rho _{p}(0)=\gamma _{p}(y)}$

due to theorem 8.2 follows ${\displaystyle \rho _{p}=\gamma _{\gamma _{p}(y)}}$  and therefore

{\displaystyle {\begin{aligned}\Phi _{\mathbf {V} }(x,\Phi _{\mathbf {V} }(y,p))&=\Phi _{\mathbf {V} }(x,\gamma _{p}(y))\\&=\gamma _{\gamma _{p}(y)}(x)\\&=\rho _{p}(x)\\&=\gamma _{p}(x+y)\\&=\Phi _{\mathbf {V} }(x+y,p)\end{aligned}}}

2. Since ${\displaystyle 0}$  is the identity element of the group ${\displaystyle (\mathbb {R} ,+)}$ , we have

${\displaystyle \Phi _{\mathbf {V} }(e,p)=\Phi _{\mathbf {V} }(0,p)=\gamma _{p}(0)=p}$ ${\displaystyle \Box }$

Theorem 9.5:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ , let ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$ , let ${\displaystyle \Phi _{\mathbf {V} }}$  be the flow of ${\displaystyle \mathbf {V} }$ , and let ${\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}$  be arbitrary. Then we have:

${\displaystyle \forall p\in M:{\mathfrak {L}}_{\mathbf {V} }\varphi (p)=\lim _{h\to 0}{\frac {\Phi _{\mathbf {V} ,h}^{*}(\varphi )(p)-\varphi (p)}{h}}}$

Proof:

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

{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\Phi _{\mathbf {V} ,h}^{*}(\varphi )(p)-\varphi (p)}{h}}&=\lim _{h\to 0}{\frac {\varphi (\Phi _{\mathbf {V} ,h}(p))-\varphi (p)}{h}}\\&=\lim _{h\to 0}{\frac {\varphi (\gamma _{p}(h))-\varphi (p)}{h}}\\&=(\gamma _{p})_{0}'(\varphi )\\&=\mathbf {V} (p)(\varphi )=:{\mathfrak {L}}_{\mathbf {V} }\varphi (p)\end{aligned}}} ${\displaystyle \Box }$

Corollary 9.6:

From the definition of ${\displaystyle {\mathfrak {L}}_{\mathbf {V} }\varphi }$ , we obtain:

${\displaystyle \forall p\in M:\mathbf {V} \varphi (p)=\lim _{h\to 0}{\frac {\Phi _{\mathbf {V} ,h}^{*}(\varphi )(p)-\varphi (p)}{h}}}$

Theorem 9.7:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$  and let ${\displaystyle \mathbf {V} ,\mathbf {W} }$  be vector fields. Then we have:

${\displaystyle \forall p\in M,\varphi \in {\mathcal {C}}^{n}(M):{\mathfrak {L}}_{\mathbf {V} }\mathbf {W} (p)(\varphi )=\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))\circ \left(\Phi _{\mathbf {V} ,h}^{-1}\right)^{*}(\varphi )-\mathbf {W} (p)(\varphi )}{h}}}$

Proof:

Let ${\displaystyle p\in M}$  and ${\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}$  be arbitrary. Then we have:

{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))\circ \left(\Phi _{\mathbf {V} ,h}^{-1}\right)^{*}(\varphi )-\mathbf {W} (p)(\varphi )}{h}}&=\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))(\varphi \circ \Phi _{\mathbf {V} ,h}^{-1})-\mathbf {W} (p)(\varphi )}{h}}\\&=\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))(\varphi \circ \Phi _{\mathbf {V} ,h}^{-1})-\mathbf {W} (p)(\varphi )}{h}}+\mathbf {W} (p)(\mathbf {V} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )\end{aligned}}}
${\displaystyle \left|\mathbf {W} (p)(\varphi )-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|\leq \left|\mathbf {W} (p)(\varphi )-\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|+\left|\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|}$

Let ${\displaystyle (O,\phi )}$  be contained in the atlas of ${\displaystyle M}$  such that ${\displaystyle p\in O}$ . We write

${\displaystyle \mathbf {W} (q)=\sum _{j=1}^{d}\mathbf {W} _{\phi ,j}(q)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{q}}$

for all ${\displaystyle q\in O}$ .

We now choose ${\displaystyle \epsilon >0}$  such that ${\displaystyle B_{\epsilon }(\phi (p))\subseteq \phi (O)}$  (which is possible since ${\displaystyle \phi (O)}$  is open as ${\displaystyle (O,\phi )}$  is in the atlas of ${\displaystyle M}$ ). If we choose ${\displaystyle h\in \gamma _{p}^{-1}()}$  we have

${\displaystyle \mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)=\sum _{j=1}^{d}\mathbf {W} _{\phi ,j}(\Phi _{\mathbf {V} ,h}^{-1}(p))\left({\frac {\partial }{\partial \phi _{j}}}\right)_{\Phi _{\mathbf {V} ,h}^{-1}(p)}}$

From theorem 5.5, we obtain that all the functions ${\displaystyle }$ are contained in ${\displaystyle {\mathcal {C}}^{\infty }(M)}$ .

Corollary 9.8:

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