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 $I_q=I_p-s$.

Fix $(\tau ,p_0)\in \mathcal{D}$. The trajectory segment $K={\gamma }_{p_0}([0,\tau ])$ if $\tau \ge 0$, or $K={\gamma }_{p_0}([\tau ,0])$ if $\tau <0$, is compact. By [L2], every $q\in K$ has an open neighbourhood $U_q$ and a smooth local flow ${\mathrm{\Psi }}_q:(-{\epsilon }_q,{\epsilon }_q)\times U_q\to M$. A finite subcover of $K$ gives a common lower time bound $\epsilon >0$ for these finitely many local flows. Partition the interval from $0$ to $\tau$ as $0=t_0,t_1,\dots ,t_m=\tau$ with $\mathrm{\mid }{t}_j-t_{j-1}\mathrm{\mid }<\epsilon$. For each $j$, choose one of the finitely many $U_q$ containing ${\gamma }_{p_0}(t_{j-1})$ and use its local flow for the time increment $t_j-t_{j-1}$. This requires only finitely many choices.

Starting with the identity near $p_0$, compose the finitely many smooth local time-increment maps from step 2.1. By continuity, shrink the initial neighbourhood $U$ of $p_0$ finitely many times so that every intermediate image lies in the domain of the next local map. The resulting smooth map $p\mapsto q_m(p)$ satisfies $q_m(p_0)={\gamma }_{p_0}(\tau )$; uniqueness of integral curves makes $q_m(p)=\mathrm{\Phi }(\tau ,p)$ for every $p\in U$. Apply [L2] once more at $q_m(p_0)$, and shrink $U$ so that $q_m(U)$ lies in its local-flow neighbourhood. For $t$ in a small open interval $J$ about $\tau$, the formula $\mathrm{\Psi }(t-\tau ,q_m(p))$ is smooth on $J\times U$ and, by uniqueness, equals $\mathrm{\Phi }(t,p)$. Hence $J\times U\subseteq \mathcal{D}$. Since $(\tau ,p_0)$ was arbitrary, $\mathcal{D}$ 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 $X$.