Differentiable Manifolds/Group actions and flows

The flow of a vector field

Theorem 9.4: Let $M$ be a manifold of class ${𝒞}^n$, where $n\in \mathbb{N}\cup \{\mathrm{\infty }\}$ ($n$ must be $\ge 1$), let $𝐕\in 𝔛(M)$ and let ${\mathrm{\Phi }}_{𝐕}$ be the flow of $𝐕$. 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 $𝐕$ is equal to $ℝ$, then the flow of $𝐕$ is a flow.

Proof:

Let $p\in M$ be arbitrary.

1.

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

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

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

$(\phi \circ {\rho }_p{)}^{\prime }(x)=(\phi \circ {\gamma }_p{)}^{\prime }(x+y)=({\gamma }_p{)}_{x^{\prime }+y}(\phi )=𝐕({\gamma }_p(x+y))(\phi )=𝐕({\rho }_p(x))(\phi )$

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

$\begin{array}{cc}{\mathrm{\Phi }}_{𝐕}(x,{\mathrm{\Phi }}_{𝐕}(y,p))& ={\mathrm{\Phi }}_{𝐕}(x,{\gamma }_p(y))\\ & ={\gamma }_{{\gamma }_p(y)}(x)\\ & ={\rho }_p(x)\\ & ={\gamma }_p(x+y)\\ & ={\mathrm{\Phi }}_{𝐕}(x+y,p)\end{array}$

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

${\mathrm{\Phi }}_{𝐕}(e,p)={\mathrm{\Phi }}_{𝐕}(0,p)={\gamma }_p(0)=p$$◻$

Proof:

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

$\begin{array}{cc}\underset{h\to 0}{\lim }\frac{{\mathrm{\Phi }}_{𝐕,h}^{*}(\phi )(p)-\phi (p)}{h}& =\underset{h\to 0}{\lim }\frac{\phi ({\mathrm{\Phi }}_{𝐕,h}(p))-\phi (p)}{h}\\ & =\underset{h\to 0}{\lim }\frac{\phi ({\gamma }_p(h))-\phi (p)}{h}\\ & =({\gamma }_p{{)}_0}^{\prime }(\phi )\\ & =𝐕(p)(\phi )=:{𝔏}_{𝐕}\phi (p)\end{array}$$◻$

Corollary 9.6:

From the definition of ${𝔏}_{𝐕}\phi$, we obtain:

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

Proof:

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

$\begin{array}{cc}\underset{h\to 0}{\lim }\frac{𝐖({\mathrm{\Phi }}_{𝐕,h}(p))\circ {\left({\mathrm{\Phi }}_{𝐕,h}^{-1}\right)}^{*}(\phi )-𝐖(p)(\phi )}{h}& =\underset{h\to 0}{\lim }\frac{𝐖({\mathrm{\Phi }}_{𝐕,h}(p))(\phi \circ {\mathrm{\Phi }}_{𝐕,h}^{-1})-𝐖(p)(\phi )}{h}\\ & =\underset{h\to 0}{\lim }\frac{𝐖({\mathrm{\Phi }}_{𝐕,h}(p))(\phi \circ {\mathrm{\Phi }}_{𝐕,h}^{-1})-𝐖(p)(\phi )}{h}+𝐖(p)(𝐕\phi )-𝐖(p)(𝐕\phi )\end{array}$

$|𝐖(p)(\phi )-𝐖({\mathrm{\Phi }}_{𝐕,h}^{-1}(p))\left(\frac{\phi \circ {\mathrm{\Phi }}_{𝐕,h}-\phi }{h}\right)|\le |𝐖(p)(\phi )-𝐖(p)\left(\frac{\phi \circ {\mathrm{\Phi }}_{𝐕,h}-\phi }{h}\right)|+|𝐖(p)\left(\frac{\phi \circ {\mathrm{\Phi }}_{𝐕,h}-\phi }{h}\right)-𝐖({\mathrm{\Phi }}_{𝐕,h}^{-1}(p))\left(\frac{\phi \circ {\mathrm{\Phi }}_{𝐕,h}-\phi }{h}\right)|$

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

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

for all $q\in O$.

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

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

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

Corollary 9.8: