---
tag: 7CJ3P443NQ
kind: statement
role: theorem
name: Ehresmann's fibration theorem
msc: [58A05, 55R10, 57R35]
uses: [F45VQ47GPV, BMY55EW3RF, 15XEBGNGHC, CX4M8K61AB, 0KZZ70ZHXP, TD6VEP3SCE, 06738GGWWK]
about: [BMY55EW3RF, CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Ehresmann%27s_lemma
same_as:
  wikidata: Q867141
  wikipedia: Ehresmann's lemma
---

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]].
Let $f \colon M \to N$ be a smooth map that is a surjective [[submersion|BMY55EW3RF]] and a [[proper map|15XEBGNGHC]].
Then $f$ is a locally trivial [[fiber bundle|CX4M8K61AB]].
That is, for every $q \in N$ there are an open neighbourhood $V$ of $q$ and a [[diffeomorphism|0KZZ70ZHXP]] $\varphi \colon f^{-1}(V) \to V \times f^{-1}(q)$ such that $\mathrm{pr}_1 \circ \varphi = f$ on $f^{-1}(V)$.

## Remarks

The idea of the proof: choose a complement of the kernel of $df$ in $TM$, for instance the orthogonal complement for a Riemannian metric; this is an [[Ehresmann connection|TD6VEP3SCE]].
Over a coordinate ball $V$ around $q$, lift the coordinate vector fields of $V$ to horizontal vector fields on $f^{-1}(V)$.
Properness makes their [[flows|06738GGWWK]] exist as long as the flow downstairs stays in $V$.
Composing these flows moves the fiber $f^{-1}(q)$ onto every nearby fiber, and this gives $\varphi$.

Example: if $M$ is compact and $N$ is connected, every submersion $f \colon M \to N$ with $M$ nonempty is proper and surjective, so all fibers of $f$ are diffeomorphic.

Properness is needed.
Let $M = \R^2 \setminus \set{0}$, $N = \R$ and $f(x, y) = x$.
Then $f$ is a surjective submersion, but its fiber over $0$ is two disjoint half-lines while its fiber over $1$ is one line.
These fibers are not diffeomorphic, so $f$ is not locally trivial near $0$.
Here $f$ is not proper: the preimage of $[-1, 1]$ is not compact.

## Explanations

### L1 human

A submersion is a smooth map with no critical points, so near each point it looks like a projection.
The theorem says that if, in addition, preimages of compact sets are compact, then the map looks like a projection near each whole fiber, not only near each point.
In particular all fibers over a connected base are the same manifold.

### L2 human

The theorem is a main source of fiber bundles in geometry and topology.
For instance, a proper family of smooth hypersurfaces with no singular member has diffeomorphic members, which is the start of the study of monodromy and of the Milnor fibration.
It is the smooth counterpart of the fact that a proper local homeomorphism between connected manifolds is a covering map.
For non-proper maps, local triviality can still hold, but one must control the behaviour at infinity by other means.