CR Geometry — Mathematics Knowledge System

Overview

CR (Cauchy-Riemann) geometry is the study of smooth manifolds equipped with a CR structure — a partial complex structure on a codimension-1 contact distribution. CR manifolds arise naturally as boundaries of complex manifolds (real hypersurfaces in ℂⁿ) and as the odd-dimensional counterpart of complex manifolds. The Tanaka-Webster connection and the associated curvature encode the intrinsic geometry of CR manifolds, while the CR equivalence problem (solved by Cartan for dim 3 and in general by Tanaka and Webster) determines when two CR manifolds are locally equivalent.

Intuition

A CR manifold is a real manifold that 'halfway' looks complex: in each tangent hyperplane there is a complex subspace (the CR structure), but the remaining direction is real. This arises concretely for hypersurfaces in ℂⁿ: the complex tangent directions at a point of the hypersurface form a complex vector space of (complex) dimension n−1, and the Levi form measures how curved the hypersurface is in the complex sense. The sphere S²ⁿ⁻¹ ⊂ ℂⁿ is the flat model of CR geometry.

Formal Definition

Definition

A CR manifold of hypersurface type is a smooth manifold M of real dimension 2n+1 with a rank-n complex sub-bundle T^{1,0}M ⊆ TM ⊗ ℂ satisfying T^{1,0}M ∩ T^{0,1}M = {0} and formal integrability.

T^{1,0}M \cap \overline{T^{1,0}M} = \{0\}

T^{1,0} and its conjugate intersect trivially

[\Gamma(T^{1,0}M),\, \Gamma(T^{1,0}M)] \subseteq \Gamma(T^{1,0}M)

CR integrability (formal integrability condition)

\mathcal{L}(X, Y) = -i\,d\theta(X, \bar{Y}), \quad X, Y \in T^{1,0}M

θ is the contact form; M is strictly pseudoconvex if L is positive definite

Levi form (measures strict pseudoconvexity)

\text{Flat model: } S^{2n-1} \subset \mathbb{C}^n, \quad T^{1,0}S^{2n-1} = TS^{2n-1} \cap T^{1,0}\mathbb{C}^n

The sphere as flat CR model

Theorems

Theorem 1: Cartan-Chern-Moser Normal Form

\text{Every strictly pseudoconvex CR hypersurface in } \mathbb{C}^{n+1} \text{ can be brought to a normal form near any point, analogous to the Darboux normal form for contact manifolds. The normal form encodes all CR invariants.}

Theorem 2: Tanaka-Webster Connection

\text{A strictly pseudoconvex CR manifold } (M, T^{1,0}M, \theta) \text{ admits a canonical connection } \nabla \text{ (the Tanaka-Webster connection) preserving both the contact structure and the CR structure, with prescribed torsion.}

Theorem 3: Embedding Problem (Boutet de Monvel)

\text{Every compact strictly pseudoconvex CR manifold of real dimension } \geq 5 \text{ can be CR-embedded into some } \mathbb{C}^N. \text{ The 3-dimensional case has obstructions.}

Worked Examples

1
View ℂ² with coordinates (z₁, z₂) = (x₁ + iy₁, x₂ + iy₂). The sphere S³ = {|z₁|² + |z₂|² = 1}. The complex tangent space T^{1,0}S³ at z is the complex hyperplane T^{1,0}_{z}ℂ² ∩ (T_zS³ ⊗ ℂ).
$S^3 = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\}$
2
T^{1,0}ℂ² is spanned by ∂/∂z₁, ∂/∂z₂. The real tangent space T_zS³ consists of vectors v with Re⟨v, z⟩ = 0 (in ℂ²). The complex tangent space: T^{1,0}_zS³ = {w ∈ ℂ² : ⟨w, z⟩_ℂ = 0} ∩ T^{1,0}ℂ². This is 1-dimensional (complex), spanned by L = z̄₂ ∂/∂z₁ − z̄₁ ∂/∂z₂.
$T^{1,0}_z S^3 = \ker(dF^{1,0}) = \{w \in \mathbb{C}^2 : \bar{z}_1 w_1 + \bar{z}_2 w_2 = 0\}$
3
The contact form on S³ is θ = i(∂̄ − ∂)|z|²/2 restricted to S³. Explicitly θ = i/2 Σⱼ (z̄ⱼ dzⱼ − zⱼ dz̄ⱼ).
$\theta = \frac{i}{2}(\bar{z}_1\, dz_1 - z_1\, d\bar{z}_1 + \bar{z}_2\, dz_2 - z_2\, d\bar{z}_2)$
4
The Levi form: L(L, L̄) = −i dθ(L, L̄). For the standard sphere in ℂ², dθ = i(dz₁ ∧ dz̄₁ + dz₂ ∧ dz̄₂) restricted to S³. Computing L(L, L̄) = |z₂|² + |z₁|² = 1 > 0.
$\mathcal{L}(L, \bar{L}) = 1 > 0$
5
The Levi form is positive definite, confirming S³ is strictly pseudoconvex. It is the flat (curvature-zero) CR model in dimension 3.
$S^3 \text{ is strictly pseudoconvex with Levi form } \mathcal{L} = 1$

✓ Answer

S³ ⊂ ℂ² carries the CR structure T^{1,0}S³ = {w : z̄·w = 0} (1-complex-dimensional). The Levi form evaluates to 1 (positive definite), making S³ strictly pseudoconvex — the flat model of 3-dimensional CR geometry.

Practice Problems

Hardfree response

Explain the difference between an integrable CR structure and a formally integrable one, and give an example of each.

Hardfree response

What is the Levi form of a real hypersurface M ⊂ ℂⁿ, and why does its signature characterize the pseudoconvexity of M?

Quiz

Summary

A CR manifold of hypersurface type has a rank-n complex sub-bundle T^{1,0}M satisfying formal integrability and T^{1,0} ∩ T^{0,1} = 0.
The Levi form L(X,Y) = −i dθ(X,Ȳ) measures pseudoconvexity; positive definite Levi form = strictly pseudoconvex.
The Tanaka-Webster connection is the canonical connection on a strictly pseudoconvex CR manifold, with associated curvature and Webster scalar curvature.
The flat model is S^{2n-1} ⊂ ℂⁿ; the Cartan-Chern-Moser normal form is the CR Darboux theorem.
CR geometry connects complex analysis (∂̄-problem), differential geometry (Tanaka-Webster curvature), and contact geometry (CR = contact + partial complex structure).