# nLab submersion

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

## 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.

Last revised on May 16, 2024 at 15:27:09. See the history of this page for a list of all contributions to it.