Ehresmann's lemma

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

1. a surjective submersion, and
2. 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