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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:
Last updated: 08-04-2005 19:41:56
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details