Examples and Definition of Foliations

Simple examples of foliations are given by submersions. A (smooth) submersion f : M → B is a map of manifolds with a surjective derivative map at every point of M. The inverse images of points in the target space form a family of closed submanifolds of M, the leaves of the foliation F on M defined by f. All these submanifolds have the same dimension p. If n denotes the dimension of M, and q the dimension of B, then n = p+q. A particularly simple situation is the case of a foliation of M by the level hypersurfaces of a smooth function f : M → ℝ. The submersion condition is the requirement of the absence of critical points for f (this is of course only possible if M is not compact).