Prove that $TS$ is subbundle of $TM|_S$ for immersed submanifold asked how to show that $TS$ is a subbundle of $TM|_S$ when $S\subseteq M$ is an immersed submanifold. However, the origin of this question is Example 10.33(c) of Professor Lee’s Introduction to Smooth Manifolds (ISM) and its associated Problem 10-14, in which both $S$ and $M$ are allowed to have nonempty boundaries. I know how to modify the proof given in the accepted answer to handle the case where $S$ has a boundary, using Theorem 5.51 in place of Theorem 5.8. But handling the case where $M$ has a boundary is proving to be difficult for me. My question is: how does one show that the example holds in the nonempty boundary of $M$ case?
What I’ve tried so far is to embed $M$ in its double $D(M)$ given in Example 9.32 and the errata. I apply the result for the no-boundary case to get a subbundle relation inside of $T(D(M))$, but I am experiencing many technical difficulties trying to bring the result back to $TS$ and $TM|_S$.
In order to avoid fooling myself, I have been working the problem making NO standard identifications. So if $i\colon S\to M$ is the inclusion map, then I’m trying to show that $di(TS)$ is a subbundle of $TM|_S$. I have (using the terminology of Theorem 9.29 (Attaching Smooth Manifolds Along Their Boundaries)) that $D(M)$ is a smooth manifold (without boundary), $M’\subseteq D(M)$ is a regular domain, and that there is a diffeomorphism $H\colon M\to M’$. I can prove that $H$ restricts to a diffeomorphism $H_{S,H(S)}\colon S\to H(S)$ and that the inclusion map $i_{H(S),M’}\colon H(S)\to M’$ is a smooth immersion. Letting $i_{M’,D(M)}\colon M’\to D(M)$ be the inclusion map, I have that $i_{M’,D(M)}$ is a smooth embedding and furthermore, for any $H(p)\in M’$, $d(i_{M’,D(M)})_{H(p)}\colon T_{H(p)}M’\to T_{H(p)}(D(M))$ is an isomorphism, due to the fact that $M’$ has codimension $0$ in $D(M)$.
The inclusion map $i_{H(S),D(M)}=i_{M’,D(M)}\circ i_{H(S),M’}\colon H(S)\to D(M)$ is then a smooth immersion, so $H(S)$ is an immersed submanifold with boundary of the smooth manifold (without boundary) $D(M)$. I can now apply the result for the boundaryless case to get that $d(i_{H(S),D(M)})(T(H(S))$ is a smooth subbundle of $T(D(M))|_{H(S)}$. Now the challenge is to pull this back to show that $di(TS)$ is a smooth subbundle of $TM|_S$.
I tried showing this directly from the definitions, but ran into roadblocks. So I decided to try to go with two applications of Lemma 10.32 (Local Frame Criterion for Subbundles). So let $p\in S$. Then $H(p)\in H(S)$, and by one application of Lemma 10.32, there exists a neighborhood $H(U)$ of $H(p)$ in $H(S)$, where $U$ is a neighborhood of $p$ in $S$, and smooth local sections $\Sigma_1,\dots,\Sigma_k\colon H(U)\to T(D(M))|_{H(S)}$ such that $\Sigma_1(H(q)),\dots,\Sigma_k(H(q))$ form a basis for $D_{H(q)}$ at each $q\in U$, where $D=d(i_{H(S),D(M)})(T(H(S)))$. Now, all I have to do is to convert this into smooth local sections $\sigma_1,\dots,\sigma_k\colon U\to TM|_S$ such that $\sigma_1(q),\dots,\sigma_k(q)$ form a basis for $\{q\}\times di_q(T_q(S))$ at each $q\in U$, and then apply Lemma 10.32 again.
I believe it isn’t too hard to specify what $\sigma_i(q)$ should equal, and it won’t be hard to show $\sigma_i$ is a local section and that $\sigma_1(q),\dots,\sigma_k(q)$ form a basis for $\{q\}\times di_q(T_q(S))$, but I will run into trouble showing that $\sigma_i$ is smooth. In preparation for this, I am able to show that the restriction $d(i_{M’,D(M)})_{TM’,T(D(M))|_{M’}}\colon TM’\to T(D(M))|_{M’}$ of $d(i_{M’,D(M)})\colon TM’\to T(D(M))$ is a diffeomorphism, and in fact a smooth bundle isomorphism. Of course, $dH\colon TM\to TM’$ is also a diffeomorphism, so I have their composition being a diffeomorphism between $TM$ and $T(D(M))|_{M’}$. Then for $q\in U$, we have that $\Sigma_i(H(q))\in T(D(M))|_{H(S)}\subseteq T(D(M))|_{M’}$, so we can apply the inverse of the above diffeomorphism to get an element of $TM$ which is in the fiber above $q$, so is in $TM|_S$. So I define \begin{equation*} \sigma_i(q) =((dH)^{-1}\circ d(i_{M’,D(M)})_{TM’,T(D(M))|_{M’}}^{-1}\circ\Sigma_i\circ H)(q), \end{equation*} for $q\in U$. Now, while this looks smooth, it doesn’t actually map where we want. That is, the given composition maps to $TM$, so we have to restrict the codomain to $TM|_S$. Unfortunately, $TM|_S$ is an immersed submanifold with or without boundary of TM, so we cannot say that the restriction would be smooth.
At this point I’ve run out of ideas.