I'm trying to prove Theorem 9.24 (Boundary Flowout Theorem) from John. M. Lee's Introduction to Smooth Manifolds . There, all steps are like proving Flowout Theorem , but in last one, I just can show that ${\Phi|_o}_{\delta}$ is an injective smooth immersion, not an embedding
(We can't use Proposition 4.22(d) beacause of boundary in M).
Also there is a hint in Problem 9.11 but I have no idea how to use it:
"define $\Phi$ first
in boundary coordinates and use uniqueness to glue together the local definitions.
To obtain an embedding, make sure $\delta(p)$ is no more than half of the
first time the integral curve starting at $p$ hits the boundary (if it ever does)."
Best Answer
The theorem that you're asking how to prove is the following (rephrased from the book):
Boundary Flowout Theorem: Suppose that $M$ is a smooth manifold with nonempty boundary and that $N$ is a smooth vector field over $M$ such that $N$ is inward-pointing at any point of the boundary $\partial M$. Then there exists a smooth positive function $\delta : \partial M\rightarrow\mathbb{R}_+$ and a smooth map $\Phi : \mathcal{P}_\delta\rightarrow M$, where $$\mathcal{P}_\delta=\left\{ \left(t,p \right) : p\in\partial M\ \ 0\le t<\delta \left(p \right)\right\},$$ such that $\Phi$ is a smooth embedding onto an open neighborhood of $\partial M$ that satisfies the following property: for any $p\in\partial M$, the curve $t\mapsto\Phi \left(t,p \right)$ for $0\le t<\delta \left(p \right)$ is the integral curve of $N$ starting at $p$.
You should follow the first half of the hint given in the book, which is done as follows:
1.) Choose boundary coordinates $ \left(U,\varphi \right)$ of $M$.
2.) Push $N$ down the image of $\varphi$ to $\mathbb{H}^n$ to get $\hat{N}=\varphi_\ast N$, and then extend that smoothly to an open subset of $\mathbb{R}^n$. Call that smoothly extended vector field ${\hat{N}}_E$ ("$E$ " for "extension").
3.) Then use the flow of ${\hat{N}}_E$ to construct a local version of $\Phi$.
4.) As a crucial step, prove that the local versions of $\Phi$ constructed as so are smooth embeddings onto open sets.
5.) Then glue these local constructions of $\Phi$ by a smooth partition of unity and prove that it satisfies the desired properties stated in the theorem.
In step 5, once you're able to prove that the constructed $\Phi$ is an injective smooth immersion, in order to prove that it's a topological embedding just note that it can be restricted to open subsets of $\mathcal{P}_\delta$ on which it's equal to the local versions of $\Phi$ that you constructed in step 3. Since you proved in step 4 that the local versions of $\Phi$ are smooth embeddings onto open sets, the same will hold for $\Phi$ on these open subsets of $\mathcal{P}_\delta$. That with the injectivity of $\Phi$ then implies that $\Phi$ is a smooth embedding onto an open subset of $M$ (which is clearly a neighborhood of $\partial M$ since $\Phi \left(0,p \right)=p$). I wrote up a full proof on my website:
https://sites.math.washington.edu/~hgrebnev/D&Writings/Z_PDF_Documents_I/Jack%20Lee%20Smooth%20Manifolds%20Notes.pdf
As is pointed out in the author's errata: the second part of the hint given in the book doesn't actually apply to this theorem because the integral curves of $N$ can never in fact hit the boundary. The reason for this is that if an integral curve of $N$ starting at a boundary point $p\in\partial M$ ever hit the boundary again, say at a point $q\in\partial M$, then $N$ would not be inward pointing at $q$.