Ehresmann's fibration theorem

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. 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 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.

Going deeper

For a student

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.