Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose derivative is everywhere surjective. Explicitly, "f" : "M" → "N" is a submersion if:$Df\_p\; :\; T\_p\; M\; o\; T\_\{f(p)\}N,$is a surjective map at every point "p" of "M". Equivalently, "f" is a submersion if it has constant rank equal to the dimension of "N"::$operatorname\{rank\},f\; =\; dim\; N.$

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).

ee also

*Immersion
*Ehresmann's fibration theorem
*Riemannian submersion

References

*J.J. Duistermaat and J.A.C. Kolk "Multidimensional Real Analysis, Differentiation", Cambridge press 2004.