Submersion (mathematics)

In the differential topology is called a differentiable map between two differentiable manifolds as submersion if its differential is surjective at any point. A special class of submersions are considered in the differential geometry of Riemannian submersions.

Points at which the differential is not surjective, it is called critical or singular.

Important example of a submersion is the projection of the first coordinate in the Euclidean space. In fact, each submersion can be represented by a suitable choice of cards locally in the form of such a projection.

If the target space, the real line, a differentiable function is exactly then a submersion if its differential vanishes nowhere identically 0.

Foliations and fiber bundles

If a submersion, then the level sets form a foliation of. This follows from the theorem of the implicit function.

If compact and a submersion, then a fiber bundle with the standard amounts as fibers. This is the statement of the theorem of Ehresmann.

An example of a submersion whose level sets form a foliation, but no fiber bundle,

The picture on the right shows the projection of this foliation, wherein the identification is used.