The fundamental theorem on flows

By [L1], for each $p\in M$ there is a unique maximal integral curve ${\gamma }_p:I_p\to M$ through $p$. Therefore the set $\mathcal{D}$ and the map $\mathrm{\Phi }(t,p)={\gamma }_p(t)$ are well defined, each fibre ${\mathcal{D}}_p=I_p$ is an open interval containing $0$, and $\mathrm{\Phi }(0,p)=p$.

Fix $p\in M$ and $s\in I_p$, and put $q:=\mathrm{\Phi }(s,p)={\gamma }_p(s)$. Then the translated curve $t\mapsto {\gamma }_p(t+s)$ is an integral curve through $q$ on the interval $I_p-s:=\{t:t+s\in I_p\}$. By uniqueness of maximal integral curves, it agrees with ${\gamma }_q$ on their common domain, so $\mathrm{\Phi }(t,\mathrm{\Phi }(s,p))=\mathrm{\Phi }(t+s,p)$ whenever both sides are defined. The same translation argument applied to ${\gamma }_q$ shows $I_q=I_p-s$.

Let $W\subseteq \mathcal{D}$ be the set of all $(t,p)\in \mathcal{D}$ such that $\mathrm{\Phi }$ is defined and smooth on some product neighbourhood $J\times U\subseteq \mathcal{D}$ of $(t,p)$. By [L2], every $(0,p)$ lies in $W$. Suppose $W\mathrm{\ne }\mathcal{D}$. Choose $(\tau ,p_0)\in \mathcal{D}\setminus W$; replacing $\tau$ by $-\tau$ if needed, we may assume $\tau >0$. Let $t_0:=\inf \{t\in \mathbb{R}:(t,p_0)\mathrm{\notin }W\}\mathrm{.}$ Then $t_0\in I_{p_0}$, because $(0,p_0)\in W$ and $I_{p_0}$ is an open interval containing $\tau$. Put $q_0:=\mathrm{\Phi }(t_0,p_0)$. Applying [L2] at $q_0$ gives $\epsilon >0$ and an open neighbourhood $U_0$ of $q_0$ such that the local flow is smooth on $(-\epsilon ,\epsilon )\times U_0$. Choose $t_1<t_0$ with $t_1+\epsilon >t_0$ and $\mathrm{\Phi }(t_1,p_0)\in U_0$. Because $(t_1,p_0)\in W$, there is a product neighbourhood $(t_1-\delta ,t_1+\delta )\times U_1\subseteq W$; shrinking $U_1$ if necessary, we may assume $\mathrm{\Phi }(\{t_1\}\times U_1)\subseteq U_0$.

Define $\tilde{\mathrm{\Phi }}(t,p):=\{\begin{array}{ll}\mathrm{\Phi }(t,p),& 0\le t<t_1,\\ \mathrm{\Psi }(t-t_1,\mathrm{\Phi }(t_1,p)),& t_1-\epsilon <t<t_1+\epsilon ,\end{array}$ where $\mathrm{\Psi }$ is the local flow from step 2.1. By step 1.2, the two formulas agree on the overlap, so $\tilde{\mathrm{\Phi }}$ is a smooth extension of $\mathrm{\Phi }$ to a product neighbourhood of $(t_0,p_0)$. This contradicts the choice of $t_0$. Therefore $W=\mathcal{D}$, so $\mathcal{D}$ is open and $\mathrm{\Phi }$ is smooth.

Step 3.1 makes each slice ${\mathcal{D}}_t:=\{p:(t,p)\in \mathcal{D}\}$ open in $M$. For $p\in {\mathcal{D}}_t$, step 1.2 gives $I_{\mathrm{\Phi }(t,p)}=I_p-t$, so $-t\in I_{\mathrm{\Phi }(t,p)}$ and hence $\mathrm{\Phi }(t,p)\in {\mathcal{D}}_{-t}$. The same step also yields $\mathrm{\Phi }(-t,\mathrm{\Phi }(t,p))=\mathrm{\Phi }(0,p)=p,$ and symmetrically $\mathrm{\Phi }(t,\mathrm{\Phi }(-t,q))=q$ for $q\in {\mathcal{D}}_{-t}$. Thus ${\mathrm{\Phi }}_t:{\mathcal{D}}_t\to {\mathcal{D}}_{-t}$ is a diffeomorphism with inverse ${\mathrm{\Phi }}_{-t}$.

Steps 1.1-4.1 show that $\mathrm{\Phi }:\mathcal{D}\to M$ is a smooth local flow whose time slices are exactly the maximal integral curves of $X$. Any other local flow of $X$ has the same time slices by uniqueness of integral curves, so its domain is contained in $\mathcal{D}$ and its map agrees with $\mathrm{\Phi }$. Therefore $\mathrm{\Phi }$ is the unique maximal local flow generated by $X$.