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

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

then it is a locally trivial fibration. This is a foundational result in differential topology due to Charles Ehresmann, and has many variants.

See also

  • 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
  • Kolář, Ivan; Michor, Peter W.; Slovák, Jan (1993). . Berlin: Springer-Verlag. ISBN3-540-56235-4. MR. Zbl.