nLab distribution of subspaces

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 MM is a real vector subbundle of the tangent bundle TMT M.

A complex distribution is a complex vector subbundle of the complexified tangent bundle T MT_{\mathbb{C}}M of MM.

A distribution of hyperplanes is a distribution of codimension 11 in TMT M; a distribution of complex hyperplanes is a distribution of complex codimension 11 in T MT_{\mathbb{C}} M.

Examples

One class of examples comes from smooth foliations by submanifolds of constant dimension m<nm\lt n. Then the tangent vectors at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension mm. 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.