# A relative version of Ehresmann's theorem

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.)

• 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." Commented Apr 25, 2020 at 21:28
• @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 $\psi (N\cap \pi^{-1}(U))=N_b\times U$. Commented Apr 26, 2020 at 4:14
• 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. Commented Apr 28, 2020 at 18:37
• @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. Commented May 1, 2020 at 2:06

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 $$(x,y)\mapsto(F_y(x),y)$$ ($$x\in M_b$$, $$y\in B$$)

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 $$x$$.