Ehresmann's lemma

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