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 exist an open neighbourhood $W\subseteq S$ of $p$, an open neighbourhood $U\subseteq M$, and $\epsilon >0$ such that $F$ restricts to a diffeomorphism from $(-\epsilon ,\epsilon )\times W$ onto its image in $U$. By [L2], after shrinking $U$ choose a local defining function $u:U\to \mathbb{R}$ for $S\cap U$. Since $X_p\mathrm{\notin }{T}_{p}S=\ker (du{)}_p$, change the sign of $u$ if needed and shrink $U,W,\epsilon$ so that $X(u)\ge c>0$ on $U$ and $F((-\epsilon ,\epsilon )\times W)\subseteq U$. This is a pointwise existence argument; it does not select such data simultaneously for all $p$.

For any local data from step 2.1 and $q\in W$, the integral curve $r\mapsto F(r,q)$ satisfies $\frac{d}{dr}u(F(r,q))=X(u)(F(r,q))\ge c$. Since $u(q)=0$, the fundamental theorem of calculus gives $\mathrm{\mid }u(F(t,q))\mathrm{\mid }\ge c\mathrm{\mid }t\mathrm{\mid }$ for $\mathrm{\mid }t\mathrm{\mid }<\epsilon$. Hence $F(t,q)\in S$ in that interval only when $t=0$.

Let $I$ be the set of all pairs $i=(W_i,{\epsilon }_i)$ where $W_i\subseteq S$ is open, ${\epsilon }_i$ is a positive rational, $(-{\epsilon }_i,{\epsilon }_i)\times W_i\subseteq \mathcal{D}$, and $F(t,q)\in S$ for $q\in W_i$, $\mathrm{\mid }t\mathrm{\mid }<{\epsilon }_i$ implies $t=0$. Steps 2.1–3.1 show that $(W_i{)}_{i\in I}$ covers $S$: each point has local data, and one can shrink its width to a positive rational. This canonical family of all admissible pairs requires no point-indexed selection. By [L3], take a locally finite subordinate partition $({\rho }_i{)}_{i\in I}$ and define $f(q)={\sum }_{i\in I}{\epsilon }_i{\rho }_i(q)$. This is smooth and positive. At each $q$, only finitely many ${\rho }_i(q)$ are nonzero; choose among them $i_0$ with maximal ${\epsilon }_i$. Then $q\in W_{i_0}$ and $f(q)\le {\epsilon }_{i_0}$ because ${\sum }_i{\rho }_i(q)=1$. Put $\delta =f\mathrm{/}2$ and $\mathcal{O}=\{(t,q)\in \mathcal{D}:\mathrm{\mid }t\mathrm{\mid }<\delta (q)\}$. This is an open neighbourhood of $\{0\}\times S$. If $F(t,q)=F(t^{\mathrm{\prime }},q^{\mathrm{\prime }})$ for two points of $\mathcal{O}$, interchange them so $f(q^{\mathrm{\prime }})\le f(q)$. The flow group law gives $F(t-t^{\mathrm{\prime }},q)=q^{\mathrm{\prime }}\in S$, while $\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }<f(q)\mathrm{/}2+f(q^{\mathrm{\prime }})\mathrm{/}2\le f(q)\le {\epsilon }_{i_0}$. The defining no-return property for $i_0$ forces $t=t^{\mathrm{\prime }}$, and then uniqueness gives $q=q^{\mathrm{\prime }}$. 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. The flow group law transports the invertible differential from $(0,q)$ in step 1.1 to every $(t,q)\in \mathcal{O}$. Thus [L4] makes $F$ a local diffeomorphism throughout $\mathcal{O}$, 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 $X$.