Main Page | See live article | Alphabetical index In mathematics, a differentiable map f from an m-manifold M to an n-manifold N is called a submersion if its differential df is an onto map at every point m of M: rk df(m) = 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. This article is a stub. You can help Wikipedia by fixing it. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.