Ehresmann's lemma

In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that a smooth mapping $f\colon M\rightarrow N$ where $M$ and $N$ are smooth manifolds such that $f$ is

a surjective submersion, and
a proper map, (in particular, this condition is always satisfied if M is compact),

is a locally trivial fibration. This is a foundational result in differential topology, and exists in many further variants. It is due to Charles Ehresmann.

References

Ehresmann, Charles (1951), "Les connexions infinitésimales dans un espace fibré différentiable", Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, pp. 29–55, MR 0042768

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