---
tag: ZC78JKZX9S
kind: concept
name: CR manifold
msc: [32V05, 32V40]
sources:
  - url: https://en.wikipedia.org/wiki/CR_manifold
same_as:
  wikidata: Q5014001
  wikipedia: CR manifold
---

## Summary

A CR manifold is a smooth real manifold with a distinguished involutive subbundle of its complexified tangent bundle, see the [[definition|A2VXXBYX6A]].
This subbundle plays the role of the holomorphic tangent directions.
The model is a real hypersurface in $\C^n$, such as the sphere $S^{2n-1}$, with the complex tangent directions that lie inside its tangent spaces.
The letters CR stand for Cauchy–Riemann.
A CR structure is what is needed to say which functions satisfy the [[tangential Cauchy–Riemann equations|X7GP9BECD8]].
CR geometry studies the boundaries of domains in [[several complex variables|BH57D2FBNR]], and its main invariant is the [[Levi form|80MTNXWW80]].

## Explanations

### L1 human

The boundary of a domain in $\C^n$ is a real manifold of odd dimension, so it cannot be a complex manifold.
Still, at each of its points some tangent directions are complex lines: those that are tangent to the boundary together with their rotation by $i$.
A CR manifold is a manifold that remembers these complex directions.
For the unit sphere in $\C^2$, the complex tangent directions at $(z_1, z_2)$ form the complex line spanned by $(-\bar z_2, \bar z_1)$.