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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." $\endgroup$ Commented Apr 25, 2020 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
    Commented Apr 26, 2020 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. $\endgroup$ Commented Apr 28, 2020 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
    Commented May 1, 2020 at 2:06

2 Answers 2

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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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I think that one can prove the relative version of Ehresmann's lemma (and also versions for manifolds with corners, which is my interest) without any technology as follows:

  1. Let us first recall the following simple proof of Ehresmann's lemma:

Let $U\subset B$ be a coordinate neighborhood $U\simeq \mathbb{R}^n$ centered at $b\in B$. Consider the coordinate vector fields $\partial_1,\dotsc,\partial_n\in \mathfrak{X}(U)$, and pick their smooth lifts $X_1,\dotsc,X_n\in\mathfrak{X}(\pi^{-1}(U))$ under $\pi$, respectively. This is possible if and only if $\pi$ is a submersion (more below). The properness of $\pi$ implies that $X_i$ are complete, so they admit complete flows $\phi_i\colon \pi^{-1}(U)\times \mathbb{R} \to \pi^{-1}(U)$. Let $f\colon \pi^{-1}(b)\times U \to \pi^{-1}(U)$ be the map defined for all $\xi\in \pi^{-1}(b)$, $t=(t_1,\dotsc,t_n)\in U$ by $$ f(\xi,t) \colon =(\phi_1^{t_1}\circ\dotsb\circ\phi_n^{t_n})(\xi). $$ We have $\pi(f(\xi,t))=t$ because $X_i$ lift $\partial_i$, and $f$ is a diffeomorphism with inverse $$ f^{-1}(x) = (t, (\phi_n^{-t_n}\circ\dotsb\circ\phi_1^{-t_1})(x)), $$ where $t=f(x)$.

  1. The relative version of Ehresmann's lemma---given a submanifold $N\subset M$ and assuming that both $\pi\colon M\to B$ and its restriction $\pi|_N\colon N\to B$ are submersions---can be obtained by choosing lifts $X_1,\dotsc,X_n$ of $\partial_1,\dotsc,\partial_n$ that are tangent to $N$ as follows:

Denote $s:=\dim(N)$, $m:=\dim(M)$. For each $x\in \pi^{-1}(U)\cap N$, let $V_x\subset M$ be a coordinate neighborhood $V_x\simeq \mathbb{R}^m$ of $x$ such that $$ V_x \cap N = \{ (x_1,\dotsc,x_m)\in V_x \mid x_{s+1}=\dotsb =x_m = 0\}. $$ The existence of such "slice charts" for a submanifold $N\subset M$ is a standard fact. Because the differential $(D\pi)_x\colon T_x N \to T_{\pi(x)}B$ is surjective, we have $n\le s$, and we can pick $n$ coordinates, w.l.o.g. $x_1,\dotsc,x_n$, such that the restriction $$ (D\pi)_x: \mathrm{span}\{\partial^x_1,\dotsc,\partial^x_n\} \to \mathrm{span}\{\partial_1,\dotsc,\partial_n\} $$ is an isomorphism. This is a simple fact from linear-algebra, namely, that any generating system contains a basis. By shrinking the neighborhood $V_x$ if necessary, we can assume that the above isomorphism holds for $(D\pi)_y$ for all $y\in V_x$. For each $y\in V_x$ we can now define the vectors $$ X_{x,i}(y) := ((D\pi)_y|_{\mathrm{span}\{\partial_1^x,\dotsc,\partial_n^x\}})^{-1}(\partial_i) \in T_y M. $$ We obtain smooth vector fields $X_{x,i}\in \mathfrak{X}(V_x)$ that lift $\partial_i$ and are tangent to $N$. We proceed similarly for $x\in \pi^{-1}(U)\backslash N$ to obtain neighborhoods $V_x$ and vector fields $X_{x,i}\in\mathfrak{X}(V_x)$ that lift $\partial_i$. Pick a cover $(V_\alpha)\subset \{ V_x \mid x\in \pi^{-1}(U)\}$ of $\pi^{-1}(U)$ with a subordinate partition of unity $\lambda_\alpha\colon V_\alpha\to [0,1]$. Define $X_i := \sum_\alpha \lambda_\alpha X_{\alpha,i}$. Then $X_i\in \mathfrak{X}(\pi^{-1}(U))$ are smooth vector fields that lift $\partial_i$, so they can be used in 1 to construct the trivializing diffeomorphism $f$. Because $X_i$ are moreover tangent to $N$, their flows through points in $N$ remain in $N$ for all times, so the diffeomorphism $f\colon \pi^{-1}(b)\times U \to \pi^{-1}(U)$ restricts to the diffeomorphism $$ f|_N\colon \pi^{-1}(b)\cap N\times U\to \pi^{-1}(U)\cap N. $$ This proves the relative Ehresmann lemma as stated in the question.

  1. Analogously, one can prove Ehresmann's lemma in the case that $M$ is a manifold with corners. Here one has to assume that the restriction of $\pi$ to any open codimension-$k$ boundary strata of $M$ is a submersion. Given $x\in M$, one picks a corner chart on $M$ that intersects only those strata that contain $x$, and one chooses the coordinates $x_1,\dotsc,x_n$, which were used to define $X_{x,i}$, in the strata of highest codimension. In this way, the vector fields $X_{x,i}$ are tangent to any strata containing $x$.
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