# Prove that $TS$ is subbundle of $TM|_S$ for immersed submanifold

I was doing Lee's smooth manifold exercise,in problem 10-14 needs to prove the following result:

Let $$S\subset M$$ be some smooth immersed submanifold,prove that $$TS$$ is subbundle of $$TM|_S$$.

It seems we need to use the local frame criterion,which needs to find a neigborhood such that exist k smooth local section $$\sigma_1,...,\sigma_k :U\to TM|_S$$ as frame for $$T_pS$$ for each $$p \in U$$.

To prove this observe that $$i:S^k\to M^n$$ is smooth immersion,I have idea how to prove it when $$S$$ is embedded submanifold,which can be shown as follows:

First for $$p\in S$$,there exist slice chart around it that is $$(U\cap S,\pi\circ\varphi = (x^1,...,x^k))$$ as chart for $$S$$,where $$(U,\varphi =(x^1,...,x^n))$$ is the corresponding chart for $$M$$.

We know chart $$(U,\varphi =(x^1,...,x^n))$$ provide a local trivialization for $$TM$$,we want to find the local section for $$TM|_S$$(needed for the local section criterion) which is obvious given by this trivialization.

Now construct the local section as follows:

$$\sigma_i : U\cup S \to (U\cap S) \times \Bbb{R}^n \to \pi^{-1}(U\cap S) \to TM|_S$$

Such that this map maps $$p \mapsto (p,e_i) \mapsto \frac{\partial}{\partial x^i}|_p$$

Our guess may be $$\sigma_1,...,\sigma_k$$ span the $$TS$$ but I can't make it clear.

Setup : Let $$\pi|_S : TM|_S \to S$$ be the ambient tangent bundle, $$\iota : S \hookrightarrow M$$ is an immersed submanifold of dimension $$k$$ of a $$n$$-dimensional smooth manifold $$M$$. For any $$p \in S$$, we have an $$k$$-dimensional subspace $$d\iota_p (T_pS) \subseteq T_pM$$. We want to show that $$\bigcup_{p \in S} d\iota_p(T_pS) \subseteq TM|_S$$ is a smooth subbundle of rank-$$k$$. To do this, it is enough to show that for any $$p \in S$$ there is a neighbourhood $$V$$ of $$p$$ and smooth local sections $$\sigma_1,\dots,\sigma_k :V \to TM|_S$$ such that at any $$q \in V$$, $$\sigma_1(q),\dots,\sigma_k(q)$$ form a basis for subspace $$d\iota_q(T_qS)$$.
Proof : Since $$S \subseteq M$$ is an immersed submanifold, then $$S$$ locally embedded. So for any $$p \in S$$, there exists a neighbourhood $$V \subseteq S$$ of $$p$$ such that $$V$$ is embedded submanifold of $$M$$. By shrinking $$V$$ if necessary, we may assume that $$V \subseteq S$$ such that $$\iota(V)=V \subseteq U$$, so $$V = V \cap U$$ is a single slice in $$U$$, where $$(U,\varphi)$$ is a slice chart for $$V$$. That is $$\varphi(V) = \{(x^1,\dots,x^n) \in \varphi(U) : x^{m+1} = \cdots= x^n = 0 \} \subseteq \varphi(U).$$ By Theorem 5.8, we have smooth (global) chart $$(V,\psi)$$ where the coordinate functions $$\psi (V) = \{ (x^1,\dots,x^m)\} \subseteq \mathbb{R}^m$$ are exactly the nonvanishing coordinates of image $$\varphi(V)$$ in $$\varphi(U)$$. Hence the representation of $$\iota : S \hookrightarrow M$$ in smooth charts $$(V,\psi)$$ and $$(U,\varphi)$$ is $$$$\tag{\star} (x^1,\dots,x^m) \mapsto (x^1,\dots,x^m,0,\dots,0).$$$$ Now for $$i=1,\dots,m$$, the coordinate vector fields $$\tau_i : V \to TS$$ defined as $$\tau_i(q) = \frac{\partial}{\partial x^i}\big|_q$$ are smooth local frame (since the component functions are constant, hence smooth) for $$TS$$ over $$V$$. Define local sections $$\sigma_i : V \to TM|_S$$ defined as $$\sigma_i(q) = d\iota_q (\tau_i(q))$$, for $$i =1,\dots,m$$, where $$d\iota_q : T_qS \to T_qM$$ is the differential of $$\iota : S \hookrightarrow M$$ at $$q\in S$$. From ($$\star$$), we can compute $$\sigma_i$$ as $$\sigma_i(q) = d\iota_q \circ \tau_i(q) = d\iota_q \Big(\frac{\partial}{\partial x^i}\bigg|_q \Big) = \frac{\partial}{\partial x^i}\bigg|_q,$$ which is form a basis for $$d\iota_q(T_qS) \subseteq T_qM$$.
We have to show that these sections are smooth sections on $$TM|_S$$. Since $$\sigma_i (V) \subseteq \bigcup_{q \in V} T_qM = (\pi|_S)^{-1}(V) = TM|_V$$ and $$TM|_V$$ is open in $$TM|_S$$, then we only need to check the smoothness in $$TM|_V$$. For the local trivialization of $$TM$$ over slice chart $$U$$ as above is $$\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^n$$, defined as $$\Phi \Big(v^i \frac{\partial}{\partial x^i}\Big|_p \Big) = (p,(v^1,\dots,v^n)).$$ So the local trivialization for $$TM|_V$$ is just the restriction of $$\Phi$$ to $$\pi^{-1}(U \cap V)$$. Denote this as $$\Psi = \Phi|_U : \pi^{-1}(U \cap V) \to (U \cap V) \times \mathbb{R}^n$$. Therefore $$\Psi \circ \sigma_i (q) =\Phi(\sigma_i(q)) =(q,e_i),$$ which is obviously smooth. Hence $$\sigma_i$$ are the smooth local sections tha we seek. Since we can do this for every point $$p \in S$$, therefore $$TS$$ identified as its image $$d\iota(TS) \subseteq TM|_S$$ is a smooth subbundle.
• Problem 10-14 actually states that both $M$ and $S$ might have nonempty boundaries, in which case slice charts are not available. Do you have a proof for this case? Commented Sep 23, 2023 at 15:59