Submersion

In mathematics, a differentiable map f : M → N from an m-manifold M to an n-manifold N is called a submersion if its differential df is a surjective map at every point p of M, or equivalently if

rank 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.

See also:

• immersion
• Ehresmann's fibration theorem