Submersion (mathematics)

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In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

Definition[edit]

Let M and N be differentiable manifolds and be a differentiable map between them. The map f is a submersion at a point if its differential

is a surjective linear map.[1] In this case p is called a regular point of the map f, otherwise, p is a critical point. A point is a regular value of f if all points p in the preimage are regular points. A differentiable map f that is a submersion at each point is called a submersion. Equivalently, f is a submersion if its differential has constant rank equal to the dimension of N.

A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal.[2] Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N then these two notions of critical point coincide. But if the dimension of M is less than the dimension of N, all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim M). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.

Submersion theorem[edit]

Given a submersion between smooth manifolds the fibers of , denoted can be equipped with the structure of a smooth manifold. This theorem coupled with the Whitney embedding theorem implies that every smooth manifold can be described as the fiber of a smooth map .

For example, consider given by The Jacobian matrix is

This has maximal rank at every point except for . Also, the fibers

are empty for , and equal to a point when . Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for .

Examples[edit]

Local normal form[edit]

If f: MN is a submersion at p and f(p) = qN, then there exists an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1, …, xm) at p and (x1, …, xn) at q such that f(U) = V, and the map f in these local coordinates is the standard projection

It follows that the full preimage f−1(q) in M of a regular value q in N under a differentiable map f: MN is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q in N if the map f is a submersion.

Topological manifold submersions[edit]

Submersions are also well-defined for general topological manifolds.[3] A topological manifold submersion is a continuous surjection f : MN such that for all p in M, for some continuous charts ψ at p and φ at f(p), the map ψ−1 ∘ f ∘ φ is equal to the projection map from Rm to Rn, where m = dim(M) ≥ n = dim(N).

See also[edit]

Notes[edit]

References[edit]

  • Arnold, Vladimir I.; Gusein-Zade, Sabir M.; Varchenko, Alexander N. (1985). Singularities of Differentiable Maps: Volume 1. Birkhäuser. ISBN 0-8176-3187-9.
  • Bruce, James W.; Giblin, Peter J. (1984). Curves and Singularities. Cambridge University Press. ISBN 0-521-42999-4. MR 0774048.
  • Crampin, Michael; Pirani, Felix Arnold Edward (1994). Applicable differential geometry. Cambridge, England: Cambridge University Press. ISBN 978-0-521-23190-9.
  • do Carmo, Manfredo Perdigao (1994). Riemannian Geometry. ISBN 978-0-8176-3490-2.
  • Frankel, Theodore (1997). The Geometry of Physics. Cambridge: Cambridge University Press. ISBN 0-521-38753-1. MR 1481707.
  • Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). Riemannian Geometry (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-20493-0.
  • Kosinski, Antoni Albert (2007) [1993]. Differential manifolds. Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8.
  • Lang, Serge (1999). Fundamentals of Differential Geometry. Graduate Texts in Mathematics. New York: Springer. ISBN 978-0-387-98593-0.
  • Sternberg, Shlomo Zvi (2012). Curvature in Mathematics and Physics. Mineola, New York: Dover Publications. ISBN 978-0-486-47855-5.