Collar neighborhood theorem

If $\mathrm{\partial }M=\mathrm{\varnothing }$, the empty map from $\mathrm{\varnothing }\times [0,1)$ is a collar. Otherwise take $X$ on $U$ from [F1]. Write $F(p,t)={\mathrm{\Phi }}_t(p)$ wherever the local forward solution remains in $U\cap M$. In a boundary chart, extend $X$ smoothly across the face to apply the ordinary local-flow theorem. At $(p,0)$, the differential of $(y,t)\mapsto F(y,t)$ has the $n-1$ face directions and the strictly inward vector $X_p$ as its columns, so it is invertible. The Euclidean inverse function theorem gives a local diffeomorphism of ambient chart neighbourhoods. Since the boundary coordinate of $F(y,t)$ vanishes for $t=0$ and has positive $t$-derivative near $(p,0)$, it has the sign of $t$ there. Thus the restriction to $t\ge 0$ is a smooth local collar with image relatively open in $M$. [F1, F2, The Euclidean inverse function theorem]

No positive-time trajectory in $U\cap M$ starting on $\mathrm{\partial }M$ can meet $\mathrm{\partial }M$ again. If $F(p,\tau )=q\in \mathrm{\partial }M$ for $\tau >0$, choose a boundary defining coordinate $r\ge 0$ near $q$. Strict inwardness gives $dr(X)>0$ on a smaller neighbourhood. For all sufficiently small $h>0$, the trajectory on $[\tau -h,\tau ]$ lies there, so the one-variable mean-value theorem gives $r(F(p,\tau -h))<r(q)=0$, a contradiction.

The flow map is injective wherever these forward trajectories are defined in $U\cap M$. If $F(p,t)=F(q,s)$ with $t\ge s$, uniqueness applied backwards along the common terminal segment of length $s$ gives $F(p,t-s)=q$. If $t>s$, this contradicts step 1.2. If $t=s$, uniqueness back to time zero gives $p=q$. This argument uses only uniqueness along already existing trajectories; it does not claim a negative-time flow inside $M$.

For $n\ge 1$, let $V_n\subseteq \mathrm{\partial }M$ be the union of all relatively open boundary sets on which the smooth forward flow exists, stays in $U\cap M$, and is smooth for times in $[0,2\mathrm{/}n]$. Smooth dependence and local existence in [F2] make $V_n$ open; as $n$ grows, the $V_n$ cover $\mathrm{\partial }M$. Apply [F3] to this countable cover, obtaining a smooth locally finite partition $({\varphi }_n{)}_{n\ge 1}$ with $\mathrm{supp}{\varphi }_n\subseteq V_n$. Set $\delta (p)={\sum }_{n\ge 1}{\varphi }_n(p)\mathrm{/}(2n)$. It is positive and smooth. At each $p$ only finitely many terms are nonzero. If $n_0$ is the least active index, then $p\in V_{n_0}$ and $0<\delta (p)\le 1\mathrm{/}(2n_0)<2\mathrm{/}{n}_0$, so $F(p,t)$ exists in $U\cap M$ for every $0\le t<\delta (p)$. No selection from an uncountable family of local widths is used.

Put $D=\{(p,t):p\in \mathrm{\partial }M,0\le t<\delta (p)\}$. The map $F:D\to M$ is smooth and injective by steps 2.1–2.2. It is a local diffeomorphism at $t=0$ by step 1.1. At $t>0$, follow a sufficiently short initial part of the trajectory from a local collar chart in step 1.1, then use the ordinary local flow near the remaining interior trajectory to transport that chart; uniqueness makes the composition equal to $F$ near $(p,t)$. Hence $F$ is a local diffeomorphism there too. Its image is relatively open in $M$ and contains $\mathrm{\partial }M$; the local inverses agree by injectivity, so $F$ is a smooth embedding onto this open neighbourhood.

The map $(p,s)\mapsto (p,s\delta (p))$ is a smooth diffeomorphism from $\mathrm{\partial }M\times [0,1)$ onto $D$, with inverse $(p,t)\mapsto (p,t\mathrm{/}\delta (p))$. Composing it with $F$ gives a smooth embedding $c(p,s)={\mathrm{\Phi }}_{s\delta (p)}(p)$ with open image and $c(p,0)=p$. By Smooth collars of a manifold boundary, this is a smooth collar.