nLab submersion

Definition

Let $X$ and $Y$ be two smooth manifolds of finite dimension and let $f : X \to Y$ be a differentiable function between them

In components, the definition of submersion reads as follows.

Definition

The function $f : X \to Y$ is called a submersion precisely if its differential $d f\colon T X \to T Y$ is for every point $x \in X$ a surjection $d f_x\colon T_x X \to T_{f(x)} Y$; hence if all points in its image are regular values.

More abstractly formulated, this means equivalently the following.

Definition

The function $f : X \to Y$ is a submersion precisely if the canonical morphism

$$T X \to X \times_Y T Y \eqqcolon f^* T Y$$

from the tangent bundle of $X$ to the pullback of the tangent bundle of $Y$ along $f$ is a surjection.

This morphism is the one induced by the universal property of the pullback from the commuting diagram

$$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y } \,.$$

In terms of coordinates, the map $f$ is a submersion at a point $p\colon X$ 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 $f(x_1,\ldots,x_n) = (x_1,\ldots,x_m)$. This definition applies to infinite-dimensional manifolds, to non-differentiable maps, even between non-differentiable manifolds.

Properties

Pullbacks

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.

Epimorphisms and coverings

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 $Diff$ as the ambient category. They appear notably in the definition of Lie groupoids.

Ehresmann’s theorem

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

Normal form

For $f : X \to Y$ a submersion, then around every point of $X$ there is an open neighbourhood on which $f$ restricts to a projection.

Characterization in infinitesimal cohesion

A smooth function $f : X \to Y$ between smooth manifolds is canonically regarded as a morphism in the cohesive (∞,1)-topos SynthDiff∞Grpd. With respect to the canonical infinitesimal neighbourhood inclusion $i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd there is a notion of formally smooth morphism in $SynthDiff\infty Grpd$.

$f$ is a submersion precisely if it is formally smooth with respect to this infinitesimal cohesion.

See the discussion at SynthDiff∞Grpd for details.

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

• William M. Boothby, Def. III 4.3 in: An introduction to differentiable manifolds and Riemannian geometry, Academic Press (1975, 1986), Elsevier (2002) [ISBN:9780121160517, pdf]

• Serge Lang, chapter XIV in: Fundamentals of differential geometry Springer (1991)

Ehresmann’s theorem is due to

• Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles (1950), 29-55.