The flowout theorem

Fix $p\in S$. Because $X_p\mathrm{\notin }{T}_{p}S$ and $S$ has codimension one, one has $T_{p}M=\mathbb{R}{X}_p\oplus T_{p}S$. The map $F(t,q)={\mathrm{\Phi }}_t(q)$ is smooth near $(0,p)$ by [L1], and its differential at $(0,p)$ sends the time direction to $X_p$ and the $S$-directions identically onto $T_{p}S$. Hence $d{F}_{(0,p)}$ is an isomorphism.

By [L4], for each $p\in S$ there are open neighbourhoods $W_p\subseteq S$ and $U_p\subseteq M$, together with ${\epsilon }_p>0$, such that $F$ restricts to a diffeomorphism from $(-{\epsilon }_p,{\epsilon }_p)\times W_p$ onto $U_p$. By [L2], after shrinking $U_p$ we may choose a local defining function $u_p:U_p\to \mathbb{R}$ for $S\cap U_p$. Since $X_p\mathrm{\notin }{T}_{p}S=\ker (d{u}_p{)}_p$, possibly replacing $u_p$ by $-u_p$ and shrinking $U_p$ again, we may assume there is $c_p>0$ with $X(u_p)\ge c_p$ on $U_p$.

If $(t,q)\in (-{\epsilon }_p,{\epsilon }_p)\times W_p$, then the curve $r\mapsto F(r,q)$ is an integral curve of $X$, so $\frac{d}{dr}{u}_p(F(r,q))=X(u_p)(F(r,q))\ge c_p$ whenever it is defined. Because $q\in S\cap U_p$, one has $u_p(q)=0$, and the fundamental theorem of calculus gives $\mathrm{\mid }{u}_p(F(t,q))\mathrm{\mid }\ge c_p\mathrm{\mid }t\mathrm{\mid }\mathrm{.}$ In particular, for such $(t,q)$ one has $F(t,q)\in S$ if and only if $t=0$.

By [L3], choose a smooth partition of unity $({\rho }_p{)}_{p\in S}$ subordinate to the open cover $(W_p{)}_{p\in S}$ of $S$, and define $f(q):={\sum }_{p\in S}{\epsilon }_p{\rho }_p(q)\mathrm{.}$ This is a smooth positive function on $S$. For each $q\in S$, pick $p_0$ with ${\rho }_{p_0}(q)>0$ and ${\epsilon }_{p_0}$ maximal among such indices. Then $q\in W_{p_0}$ and $f(q)={\sum }_p{\epsilon }_p{\rho }_p(q)\le {\epsilon }_{p_0}{\sum }_p{\rho }_p(q)={\epsilon }_{p_0}\mathrm{.}$ Set $\delta :=f\mathrm{/}2$, and let $\mathcal{O}:=\{(t,q)\in \mathbb{R}\times S:(t,q)\in \mathcal{D},\mathrm{\mid }t\mathrm{\mid }<\delta (q)\}\mathrm{.}$ Then $\mathcal{O}$ is an open neighbourhood of $\{0\}\times S$ in $\mathbb{R}\times S$. Suppose $F(t,q)=F(t^{\mathrm{\prime }},q^{\mathrm{\prime }})$ with $(t,q),(t^{\mathrm{\prime }},q^{\mathrm{\prime }})\in \mathcal{O}$. Renaming if necessary, assume $f(q^{\mathrm{\prime }})\le f(q)$. By the group law from [L1], one has $F(t-t^{\mathrm{\prime }},q)=q^{\mathrm{\prime }}\in S$. Also $\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid }+\mathrm{\mid }{t}^{\mathrm{\prime }}\mathrm{\mid }<\frac{f(q)}{2}+\frac{f(q^{\mathrm{\prime }})}{2}\le f(q)\le {\epsilon }_{p_0}\mathrm{.}$ Step 3.1 applied inside $(-{\epsilon }_{p_0},{\epsilon }_{p_0})\times W_{p_0}$ therefore forces $t-t^{\mathrm{\prime }}=0$, and then $q=q^{\mathrm{\prime }}$. Thus $F$ is injective on $\mathcal{O}$.

Because $S$ has codimension one, the source $\mathcal{O}\subseteq \mathbb{R}\times S$ and the target $M$ have the same dimension. Step 1.1 and [L4] therefore make $F$ a local diffeomorphism at every point of $\mathcal{O}$, and step 4.1 makes it injective. Hence $F:\mathcal{O}\to M$ is a diffeomorphism onto the open submanifold $F[\mathcal{O}]$. By definition, this image is a flowout of $S$ along $X$.