Submersion (mathematics)

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.

DefinitionEdit

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 theoremEdit

Given a submersion between smooth manifolds   of dimensions   and  , for each   there are surjective charts   of   around  , and   of   around  , such that   restricts to a submersion   which, when expressed in coordinates as  , becomes an ordinary orthogonal projection. As an application, for each   the corresponding fiber of  , denoted   can be equipped with the structure of a smooth submanifold of   whose dimension is equal to the difference of the dimensions of   and  .

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  .

ExamplesEdit

Maps between spheresEdit

One large class of examples of submersions are submersions between spheres of higher dimension, such as

 

whose fibers have dimension  . This is because the fibers (inverse images of elements  ) are smooth manifolds of dimension  . Then, if we take a path

 

and take the pullback

 

we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups   are intimately related to the stable homotopy groups.

Families of algebraic varietiesEdit

Another large class of submersions are given by families of algebraic varieties   whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstauss family   of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by

 

where   is the affine line and   is the affine plane. Since we are considering complex varieties, these are equivalently the spaces   of the complex line and the complex plane. Note that we should actually remove the points   because there are singularities (since there is a double root).

Local normal formEdit

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 submersionsEdit

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 alsoEdit

NotesEdit

ReferencesEdit

  • 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.