# nLab distribution of subspaces

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)

This entry is about the concept in differential geometry and Lie theory. For the concept in functional analysis see at distribution.

# Contents

## Definition

A real distribution on a real smooth manifold $M$ is a real vector subbundle of the tangent bundle $T M$.

A complex distribution is a complex vector subbundle of the complexified tangent bundle $T_{\mathbb{C}}M$ of $M$.

A distribution of hyperplanes is a distribution of codimension $1$ in $T M$; a distribution of complex hyperplanes is a distribution of complex codimension $1$ in $T_{\mathbb{C}} M$.

## Examples

One class of examples comes from smooth foliations by submanifolds of constant dimension $m\lt n$. Then the tangent vectors at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension $m$. The distributions of that form are said to be integrable.

say something about the Frobenius theorem

## References

Discussion in the context of geometric quantization:

• N. M. J. Woodhouse, Geometric quantization

Last revised on May 5, 2023 at 05:11:01. See the history of this page for a list of all contributions to it.