nLab
submersion

Definition

A differentiable map $f:X\to Y$ between two finite dimensional manifolds is called a submersion precisely if its differential $df:TX\to TY$ is for every point $x\in X$ a surjection $d{f}_x:T_{x}X\to T_{f(x)}Y$.

In terms of coordinates, the map $f$ is a submersion at a point $p:X$ if and only if there exists a coordinate chart on $X$ near $p$ and a coordinate chart on $Y$ near $f(p)$ relative to which $f$ is the projection $f(x_1,\dots ,x_n)=(x_1,\dots ,x_m)$. This definition applies to infinite-dimensional manifolds, to non-differentiable maps, even between non-differentiable manifolds.

Properties

While the category Diff of (finite dimensional) smooth manifolds does not have all pullbacks, the pullback along a submersion always exists. This is because a submersion is transversal to every other smooth map into its codomain. Moreover, submersions are stable under pullback.

The surjective submersions (that is the submersions that are also epimorphisms in Diff) are regular epimorphisms.

Surjective submersions form a singleton Grothendieck pretopology on Diff, and so may be used in internal category theory when using $\mathrm{Diff}$ as the ambient category. They appear notably in the definition of Lie groupoids.

Ehresmann's theorem states that a proper submersion is a locally trivial fibration.

Variants

The algebraic geometry analogue of a submersion is a smooth morphism.

The analogue between arbitrary topological spaces (not manifolds) is simply an open map. There is also topological submersion, of which there are two versions.

Related concepts

• immersion, submersion, local diffeomorphism

References

For instance chapter XIV

• Serge Lang, Fundamentals of differential geometry Springer (1991)