Differentiable Manifolds/Group actions and flows

The flow of a vector field
Theorem 9.4: Let $$M$$ be a manifold of class $$\mathcal C^n$$, where $$n \in \mathbb N \cup \{\infty\}$$ ($$n$$ must be $$\ge 1$$), 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{align}

\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{align}$$

2. Since $$0$$ is the identity element of the group $$(\mathbb R, +)$$, we have
 * $$\Phi_{\mathbf V}(e, p)= \Phi_{\mathbf V}(0, p) = \gamma_p(0) = p$$

Proof:

Let $$p \in M$$ be arbitrary. We have:
 * $$\begin{align}

\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{align}$$

Corollary 9.6:

From the definition of $$\mathfrak L_{\mathbf V} \varphi$$, we obtain:
 * $$\forall p \in M : \mathbf V \varphi (p) = \lim_{h \to 0} \frac{\Phi_{\mathbf V, h}^*(\varphi)(p) - \varphi(p)}{h}$$

Proof:

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

\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{align}$$


 * $$\left| \mathbf W(p)(\varphi) - \mathbf W \left( \Phi_{\mathbf V, h}^{-1}(p) \right) \left( \frac{\varphi \circ \Phi_{\mathbf V, h} - \varphi}{h} \right) \right| \le \left| \mathbf W(p)(\varphi) - \mathbf W(p) \left( \frac{\varphi \circ \Phi_{\mathbf V, h} - \varphi}{h} \right) \right| + \left| \mathbf W(p) \left( \frac{\varphi \circ \Phi_{\mathbf V, h} - \varphi}{h} \right) - \mathbf W \left( \Phi_{\mathbf V, h}^{-1}(p) \right) \left( \frac{\varphi \circ \Phi_{\mathbf V, h} - \varphi}{h} \right) \right|$$

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 $$$$ are contained in $$\mathcal C^\infty(M)$$.

Corollary 9.8: