In mathematics, a differentiable map f from an m-manifoldM to an n-manifold N is called a submersion if its differential df is an onto map at every point p of M:
rk df(p) = dim N.
Examples include the projections in smooth vector bundles; and more general smooth fibrations. Therefore one can regard the submersion condition as a necessary condition for a local trivialization to exist. There are some converse results.
The points at which f fails to be a submersion are the critical points of f: they are those at which the Jacobian matrix of f, with respect to local coordinates, is not of maximum rank. They are the basic objects of study in singularity theory; and also in Morse theory.
