# Construct the restricted bundle $E|_S$ for immersed submanifold

Let $$S\to M$$ be the immersed submanifold. $$E$$ be the smooth vector bundle over $$M$$.We need to construct $$TM|_S$$

We can construct the restriction of $$E$$ over $$S$$ as follows(which is given in Lee's smooth manifold book page 255 as explained in the following link here)

If $$S$$ is merely immersed, we give $$\left.E\right|_{S}$$ a topology and smooth structure making it into a smooth rank- $$k$$ vector bundle over $$S$$ as follows: For each $$p \in S$$, choose a neighborhood $$U$$ of $$p$$ in $$M$$ over which there is a local trivialization $$\Phi$$ of $$E$$, and a neighborhood $$V$$ of $$p$$ in $$S$$ that is embedded in $$M$$ and contained in $$U$$. Then the restriction of $$\Phi$$ to $$\pi^{-1}(V)$$ is a bijection from $$\pi^{-1}(V)$$ to $$V \times \mathbb{R}^{k}$$, and we can apply the chart lemma to these bijections to yield the desired structure."

The question is why we need to put the neighborhood $$V$$ locally embedded in $$M$$ first. It seems slice chart lemma holds without this additional step?(To make it more clear,if embedding we may just take the local trivialization domain as $$U\cap S$$, but for immersed submanifold as shown above ,we need to shrink it to some $$V\subset U$$ such that $$V\to M$$ is local embedding.)

• I can't really understand your question. But maybe the answer is because slice chart lemma only applies to embedded submanifold? Apr 30 at 17:26
• I still think that I have some subtle point that I missed May 1 at 1:54
• A similar question see here:math.stackexchange.com/q/3233772/360262 May 1 at 1:58

To make the solution to this question clear,I will explain with example:

Just consider the $$S$$ be figure eight curve in $$M = \Bbb{R}^2$$.$$E$$ over $$M$$ just take rank-0 vector bundle $$E= \Bbb{R}^2$$.where $$\pi$$ associated to it is identity map. $$E|_S$$ as a set is just figure eight curve.

Let's see if we only use the construction same as embedded submanifold what will happen for $$E_S$$

that is construct the local trivialization as $$\Phi_U:\pi_{S}^{-1}(U\cap S) \to (U\cap S) \times \{0\}$$.We can see the local trivialization here is just all the identity $$\Phi_U:U\cap S \to U\cap S$$.

Recall we use this $$\varphi\circ\Phi_U$$ as a local chart for $$E|_S$$ in slice chart lemma. And $$U\cap S$$ should be the form of local chart for $$S$$.But $$S$$ is immersion $$U\cap S$$ is not local chart for $$S$$(near origin)

In order to solve this problem,we need to consider to shrink it into some locally embedded neiborhood.as explained in the link.