is a
surjectivelinear 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
Given a submersion between smooth manifolds of dimensions and , for each there are
surjectivecharts 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 .
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.
Maps between spheres
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
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 Weierstrass 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 form
If f: M → N is a submersion at p and f(p) = q ∈ N, then there exists an
open neighborhoodU 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: 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 in N if the map f is a submersion.
Topological manifold submersions
Submersions are also well-defined for general
topological manifolds.[3] A topological manifold submersion is a
continuous surjection f : M → N 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).