nLab
Ehresmann's theorem

Ehresmann’s theorem states that a proper submersion $f:X\to Y$ is a locally trivial fibration.

This is important in algebraic geometry because it implies that the higher direct images $R^i{f}_{*}\underline{ℂ}$ of the constant sheaf $\underline{ℂ}$ on $X$ are ($ℂ$-)local systems on $Y$. (If we work in the algebraic category, then instead of the constant sheaf $\underline{ℂ}$ we take the de Rham complex ${\Omega }_X^{•}$ and instead of the higher direct images we take the hyper-higher direct images.) The corresponding vector bundle then has a canonical flat connection, known as the Gauss-Manin connection. This is the typical setup one considers when studying variations of Hodge structure.