Submersion

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

The general definition of a submersion applies to arbitrary differentiable manifolds, but in many interesting examples these manifolds are open subsets of Euclidean spaces,

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

is a surjective linear map. In this case p is called a regular point of the map f, otherwise, p is a singular point. A point q ∈ N is a regular value of f if all points p in the pre-image f⁻¹(q) 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 Df_p has constant rank equal to the dimension of N.

If the dimension of M is greater than or equal to the dimension of N then the points at which Df_p fails to be a surjection are the critical points of f, because the rank of the Jacobian matrix of f at p is not maximal. Thus in this case, the notions of a singular point and a critical point coincide. In the opposite case, when the dimension of M is less than the dimension of N, all points are singular. Critical points are the basic objects of study in Morse theory (N = R, f is a real-valued function) and the singularity theory of differentiable maps (in general).

Examples

• Local diffeomorphisms

• Riemannian submersions

• The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.

Local normal form

If f: M → N is a submersion at p and f(p) = q ∈ N then there exist an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x₁,…,x_m) at p and (x₁,…,x_n) at q such that f(U) = V and the map f in these local coordinates is the standard projection

It follows that the full pre-image f⁻¹(q) in M of a regular value q ∈ N under a differentiable map f: M → N 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. In particular, the conclusion holds for all q ∈ N if the map f is a submersion.

See also

• Ehresmann's fibration theorem

References

• Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985), Singularities of Differentiable Maps: Volume 1, Birkhäuser, ISBN 0817631879 

• Bruce, J. W.; Giblin, P. J. (1984), Curves and Singularities, Cambridge University Press, ISBN 0521429994