nLab
CR manifold

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.

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

• HR manifold

• BR manifold?

References

The original article is

• Shiing-shen Chern, Jürgen Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1975), 219–271 (MathNet)

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)