In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion.

Definition

Let M and N be differentiable manifolds 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 is a regular value of if all points in the preimage are regular points. A differentiable map is called a submersion. Equivalently, has constant rank equal to the dimension of .

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. However, 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 one, e.g., in the formulation of Sard's theorem.

Submersion theorem

there exist surjective charts, becomes an ordinary orthogonal projection.

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

. The Jacobian matrix is

. Also, the fibers

are empty.

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.

Maps between spheres

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

. Then, if we take a path

and take the pullback

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

Families of algebraic varieties

Another large class of submersions is given by families of algebraic varieties 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 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 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