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 $f\colon M \rightarrow N$ , where $M$ and $N$ are smooth manifolds, is

a surjective submersion, and
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.

References

{{citation|last=Ehresmann|first= Charles|authorlink=Charles Ehresmann| contribution=Les connexions infinitésimales dans un espace fibré différentiable| title=Colloque de topologie (espaces fibrés), Bruxelles, 1950|publisher= Georges Thone, Liège; Masson et Cie., Paris|year= 1951 |pages= 29-55|mr=0042768}}
{{cite book|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter W.|last3=Slovák|first3=Jan|title=Natural operations in differential geometry|publisher=Springer-Verlag|location=Berlin|year=1993|isbn=3-540-56235-4|mr=1202431|zbl=0782.53013|url=https://www.emis.de///monographs/KSM/}}