# nLab CR manifold

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

tangent cohesion

differential cohesion

graded differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

A CR manifold consists of a differentiable manifold $M$ together with a subbundle $L$ of the complexified tangent bundle, $L \subset TM \otimes_\mathbf{R} \mathbf{C}$ such that $[L, L ] \subset L$ and $L \cap\overline{L} =\{ 0 \}$.

### As first-order integrable $G$-structure

CR manifold structure are equivalently certain first-order integrable G-structures (Dragomi-Tomassini 06, section 1.6), a type of parabolic geometry.

## Properties

### Relation to solutions in supergravity

A close analogy between CR geometry and supergravity superspacetimes (as both being torsion-ful integrable G-structures) is pointed out in (Lott 01 exposition (4.2)).

## Other Clifford-type Hypersurfaces

The original article is

Surveys inclides

• Wikipedia, CR manifold

• Sorin Dragomir, Giuseppe Tomassini, Differential Geometry and Analysis on CR Manifolds, Birkhäuser, 2006

Discussion from the point of view of Cartan geometry/parabolic geometry includes

• Felipe Leitner, section I.10 of Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry, 2007 (pdf)

Discussion of spherical CR manifolds locally modeled on the Heisenberg group is in:

• Robert R. Miner, Quasiconformal equivalence of spherical manifolds, Annales Academiae Scientiarium Fennicae, Series A. I. Mathematica, Volumen 19, 1994, 83-93 (pdf)