What are the sufficient and necessary conditions for surjective submersions to be locally trivial

Question

Let $E$, $M$ be smooth finite dimensional, Hausdorff and second-countable manifolds. Let $\pi:E \longrightarrow M$ be a surjective submersion.

$\pi$ is locally trivial if $\forall p\in M$, $\exists U \ni p$ open neighborhood such that there is a diffeomorphism: $$ \phi:\pi^{-1}(U)\longrightarrow U \times \pi^{-1}(p)$$ $$ \text{proj}_2\circ \phi=\pi $$

I read that if $\pi$ is proper then is locally trivial (Ehresmann theorem). But not every locally trivial submersion is proper (vector bundles for example).

Is there any sufficient and necessary condition for a surjective submersion to be locally trivial?

Answer 1

A connection for a (surjective) submersion $\pi\colon E\to M$ is a complementary subbundle $\mathcal H E\subset TE$ of the vertical bundle $\mathcal VE=ker (D\pi).$ A connection locally defines a parallel transport in $E$ along curves $\gamma\colon I\to M$, by lifting to a horizontal curve, i.e., $$\hat\gamma\colon \tilde I\to E;\; \hat\gamma'(t)\in\mathcal HE_{\hat\gamma(t)}.$$ As opposed to the case of linear connections on vector bundles (or principal connections on principal bundles), the parallel transport does not always exist globally on the interval of definition $T$ for $\gamma$, but only exists on relatively open sub-intervals $\tilde I\subset I.$ A connection is called complete if every curve $\gamma$ admits a global horizontal lift. I think the following observation can be attributed to Ehresmann: A surjective submersion is locally trivial if and only if it admits for all $p\in M$ an open neighbourhood $U$ of $p$ and a connection $\mathcal HE$ which is complete whence restricted to $U\subset M$. In fact, you can use the parallel transport to construct diffeomorpisms $\phi$ as wanted.

Answer 2

The following differential-topological characterization is Theorem B in Meigniez, Gaël. Submersions, fibrations and bundles. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771--3787. It is currently available in the website of the author: http://web.univ-ubs.fr/lmba/meigniez/docu/preprints/sfb.pdf (the page is http://web.univ-ubs.fr/lmba/meigniez/docu/travaux.html, in case the link to the pdf changes at some point). I do not really know the proof, as I only skimmed through the paper some years ago, but I remember that the paper looked very interesting to me.

Let the dimension of $M$ be $m$ and let the dimension of $E$ be $n+m$.

Theorem A surjective smooth submersion $\pi:E\to M$ is a (locally trivial fibre) bundle if and only if it admits an exhaustive, isotopy invariant, $(m−1)$-fibred family of vertical domains.

To understand the statement, we need some definitions (taken from Section II.1 of the same reference):

• A vertical domain is an $n$-dimensional compact submanifold of a fibre, $X\subset E_p$, with a smooth boundary.

Let $VE(X,E)$ denote the space of vertical embeddings of $X$ into $E$ (so, those with image in a fibre), with the topology of smooth uniform convergence. Let $VE^0(X,E)$ denote the connected component of $VE(X,E)$ containing the original inclusion $X\to E_p$.

Let $VD=\coprod_{p\in M} VD_p$ be a family of vertical domains (Note that each $VD_p$ is itself a collection of vertical domains, but all in the same fibre $E_p$). It is called:

• exhaustive if every compact subset of every fibre is contained in some $X\in VD$;

• isotopy invariant if for every $X\in VD$ and every $\phi \in VE^0(X,E)$ we have $\phi(X)\in VD$;

• $r-$fibred if, for any two domains $X,X′\in VD_p$ such that $X\subset Int(X′)$, the restriction map

$$\rho_{X,X′}:VE^0(X′,E)\to VE^0(X,E)$$ is an $r$-fibration (i.e., has the homotopy lifting property for polytopes of dimension at most $r$).

In the same paper there is discussion of some other sufficient conditions for local triviality, see II.1,applications, and also on conditions for a surjective submersion to be a fibration, i.e., satisfying homotopy lifting properties (part I of the paper). For example, the following easier (i.e. with conditions possibly easier to verify) characterization is Corollary 28 in the paper:

Corollary (dim-1 base principle for bundles)

a) A surjective submersion $\pi:E→M=R^m$ is a bundle if and only if it is a bundle over each straight line in the base parallel to one of the axes.

b) A surjective submersion $\pi:E→M$ is a bundle if and only if for every smooth path $\gamma:[0,1]→M$, the pullback $\gamma^*\pi:\gamma^*E→[0,1]$ is a bundle.

c) The theorem is still true if we change "$(m−1)$-fibred" to "$0$-fibred".