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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.)

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

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