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Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.

Let $N\subset M$ be a pair of the smooth manifolds and $\pi:M\to B$ a proper submersion to a smooth base. Assume moreover that the restriction $\pi_{|N}:N\to B$ is also proper and submersive, then both $\pi$ and $\pi_{|N}$ are locally trivial by applying Ehresmann's theorem separately, but I'm wondering if they can be simultaneously locally trivialized.

More precisely, I'm looking for a proof of the following argument:

Claim: For each $b\in B$, there is a neighborhood $U$ of $b$, and a fiber preserving diffeomorphism $$\psi:\pi^{-1}(U)\xrightarrow{\cong} M_b\times U,$$ satisfying $$\psi(N\cap \pi^{-1}(U))=N_b\times U$$ where $M_b=\pi^{-1}(b)$ and $N_b=\pi_{|N}^{-1}(b)$.

The example that I have in mind is a smooth family of pairs (cubic surface, cubic curve). Explicitly take a general pencil $\mathbb P^1$ of hyperplane sections of a smooth cubic threefold $X$ over $\mathbb C$ with base locus a cubic curve $E$ (as the intersection of any two hyperplane section in the pencil). Project the incidence variety $I=\{(y,t)|y\in X\cap H_t\}\subset X\times \mathbb P^1$ to the second factor produces a family of cubic surfaces. Now $M\to B$ will be restriction of $I\to \mathbb P^1$ to the locus where cubic surfaces are smooth and $N:=E\times B$.

(For a smooth family of compact complex manifolds, one can even ask for transversely holomorphic trivialization, see page 2 in this notes, I would also like to know if it works for a pair.)

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    $\begingroup$ There is a general theorem that goes under the name of "Thom's first isotopy lemma." You can read about it in Mather's "Notes on topological stability." $\endgroup$ – Phil Tosteson Apr 25 at 21:28
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    $\begingroup$ @PhilTosteson Thanks for your comment, I checked Thom's first isotopy lemma in Mather's notes. However, I'm not sure how this lemma is related to my question since I already assumed both $\pi$ and $\pi_{|N}$ are proper and submersive, so Ehresmann's theorem applied to $M$ and $N$ separately tell us they are both locally trivial over $B$, but the argument that I am looking for is the existence of a (local) trivialization $\psi: \pi^{-1}(U)\xrightarrow{\cong} M_b\times U$ which satisfies $\psi (N\cap \pi^{-1}(U))=N_b\times U$. $\endgroup$ – AG learner Apr 26 at 4:14
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    $\begingroup$ Sorry, you need the "second isotopy lemma" to see your desired outcome as a direct consequence. However, I think that the proof of the first isotopy lemma involves proving a version of your statement. $\endgroup$ – Phil Tosteson Apr 28 at 18:37
  • $\begingroup$ @PhilTosteson Thank you! It makes more sense to me now. I hope I can dig an elementary proof out of it, since I am working on smooth submanifold case compared to the general stratified subsets considered in the first/second isotopy lemma. $\endgroup$ – AG learner May 1 at 2:06
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The answer is positive. There are several proofs of Eheresmann's genuine lemma; I think that each of them can be straightforwardly generalized and gives your relative version. But you can also, alternatively, deduce the relative version from the absolute one together with a classical theorem of Cerf, as follows. After restricting $B$ to a small compact ball centered at a point $b$, by a first application of the genuine Ehresmann lemma to $(M,\pi)$, you can assume without loss of generality that $M=M_b\times B$ and that $\pi$ is the second projection. Then, after restricting $B$ to a smaller compact ball centered at $b$, by a second application of the genuine Ehresmann lemma to $(N,\pi\vert N)$, there is a local trivialization of $\pi\vert N$ over $B$, in other words a diffeomorphism $$\psi_N:N\to N_b\times B$$ such that $\pi\circ\psi_N=\pi\vert N$. You can also see $\psi_N$ as a parametric family of smooth embeddings $$f_y:N_b\hookrightarrow M_b$$ parametrized by $y\in B$, namely $$(f_y(x),y)=\psi_N^{-1}(x,y)$$ It is classical (Cerf) that a parametric family of embeddings of $N_b$ in $M_b$ extends to a parametric family of self-diffeomorphisms (isotopies) of the ambiant manifold $M_b$: there is a smooth family $(F_y)$ of self-diffeomorphisms of $M_b$ such that $f_y=F_y\vert N_b$; and $F_b$ is the identity. Finally, the trivialization $\psi$ that you are looking for is just the inverse of $$(x,y)\mapsto(F_y(x),y)$$ ($x\in M_b$, $y\in B$)

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