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  be a differentiable map between them. The map  is a submersion at a point  if its differential

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

A word of warning: some authors use the term critical point to  describe a point where the rank of the Jacobian matrix of  at  is not maximal. Indeed, this is the more useful notion in singularity theory. If the dimension of  is greater than or equal to the dimension of  then these two notions of critical point coincide. But if the dimension of  is less than the dimension of , 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 ). 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 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 .

The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).

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 
 Any projection 
 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.

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

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 varieties 
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 bywhere  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  is a submersion at  and , then there exists an open neighborhood  of  in , an open neighborhood  of  in , and local coordinates  at  and  at  such that , and the map  in these local coordinates is the standard projection
 

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

Topological manifold submersions 
Submersions are also well-defined for general topological manifolds. A topological manifold submersion is a continuous surjection  such that for all  in , for some continuous charts  at  and  at , the map  is equal to the projection map from  to , where .

See also 
 Ehresmann's fibration theorem

Notes

References

Further reading 
https://mathoverflow.net/questions/376129/what-are-the-sufficient-and-necessary-conditions-for-surjective-submersions-to-b?rq=1

Maps of manifolds
Smooth functions