Ehresmann's lemma

Short description: On when a smooth map between smooth manifolds is a locally trivial fibration

In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping [math]\displaystyle{ f\colon M \rightarrow N }[/math], where [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] 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

• Thom's first isotopy lemma

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 
• Kolář, Ivan; Michor, Peter W.; Slovák, Jan (1993). Natural operations in differential geometry. Berlin: Springer-Verlag. ISBN 3-540-56235-4. https://www.emis.de///monographs/KSM/.