Submersions

A submersion is a smooth map f: M → N between smooth manifolds whose differential dfp is surjective at every point p in M. This implies that the dimension of N is at most the dimension of M, and, by the constant rank theorem, in suitable local coordinates f resembles a projection onto the first dim N coordinates.

Consequences follow from the surjectivity of the differential. If y ∈ N is a regular value (dfx is surjective for all x with f(x) = y), then the preimage f⁻¹(y) is a smooth submanifold of M of dimension dim M − dim N. For a submersion, every point of the image is a regular value.

If f is furthermore a proper map, Ehresmann’s theorem says that f is a locally trivial fibration: around each point of N, M looks like a product U × F with F = f⁻¹(n0) for some n0 ∈ U, and the projection corresponds to f. Thus submersions organize spaces into fiber bundles in many contexts.

Common examples include the projection map π: M × N → M, which is a submersion, and quotient maps arising from smooth, proper group actions, where the fibers are the orbits of the action. Submersions are dual to immersions: while immersions have injective differentials, submersions have surjective differentials. The submersion condition is central to differential topology and the theory of fiber bundles.