nLab CR manifold

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)

Contents

Definition

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

As first-order integrable GG-structure

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

Examples

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

References

The original article is

Surveys:

See also

Discussion of orbifolds with CR-structure:

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)

Last revised on July 18, 2020 at 18:45:05. See the history of this page for a list of all contributions to it.