Submersion (mathematics)

"Regular point" redirects here. For "regular point of an algebraic variety", see Singular point of an algebraic variety.

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

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

$Df_p : T_p M \to T_{f(p)}N\,$

is a surjective linear map. In this case p is called a regular point of the map f, otherwise, p is a critical 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.

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. 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 more commonly used, e.g. in the formulation of Sard's theorem.

Examples

• Any projection $\pi:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^n\subset\mathbb{R}^{m+n}$

• 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

$f(x_1, \ldots, x_n, x_{n+1}, \ldots, x_m) = (x_1, \ldots, x_n).$

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 (also known as the submersion 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 0-8176-3187-9 

• Bruce, J. W.; Giblin, P. J. (1984), Curves and Singularities, Cambridge University Press, ISBN 0-521-42999-4