Differentiable Manifolds/Group actions and flows

Definitions of group actions and flows

Definition 9.1:

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

• $(gh)*x=g*(h*x)$
• $e*x=x$

Definition 9.2:

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

The flow of a vector field

Definition 9.3:

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

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

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

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

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

Proof:

Let $p\in M$ be arbitrary.

1.

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

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

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

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

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

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

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

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

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

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

Theorem 9.5:

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

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

Proof:

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

${\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}}$$\Box$

Corollary 9.6:

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

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

Theorem 9.7:

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

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

Proof:

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

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

$\left|\mathbf {W} (p)(\varphi )-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|\leq \left|\mathbf {W} (p)(\varphi )-\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|+\left|\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|$

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

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

for all $q\in O$.

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

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

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

Corollary 9.8: