Submersion (mathematics)
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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 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. (However, in Morse theory, critical point means that the derivative is actually zero, so that at some points a function may be neither a submersion nor a critical point in the Morse theoretic sense).
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