CR geometry

CR geometry begins with a complex subbundle of the complexified tangent bundle that behaves like the antiholomorphic or holomorphic tangent directions of a complex manifold. A CR manifold of CR dimension n and codimension k has a complex rank-n distribution, usually denoted T^1,0, together with its conjugate T^0,1; formal integrability requires sections of T^1,0 to be closed under the Lie bracket. The standard examples are real hypersurfaces in complex manifolds, where the CR distribution is the complex tangent space. The unit sphere in complex Euclidean space is the model strongly pseudoconvex example, while boundaries of pseudoconvex domains supply much of the subject’s analytic motivation.¹

When the CR codimension is one, a contact form often identifies the transverse real direction. This setting supports analogues of Riemannian and Kähler constructions, but the geometry is intrinsically sub-Riemannian and depends on the chosen contact or pseudo-Hermitian data.

The Levi form is the central local invariant distinguishing CR geometries with different analytic behavior. For a contact form, it is obtained from the exterior derivative restricted to the complex distribution and is a Hermitian form; its signature defines strict pseudoconvexity, Levi indefiniteness, or Levi degeneracy. Strong pseudoconvexity is especially important because it supplies positivity for several fundamental operators and estimates.

The tangential Cauchy–Riemann operators act along the CR directions, and their associated Kohn Laplacian governs boundary versions of the complex Green operator and the ∂̄-Neumann problem.² Questions about closed range, regularity, cohomology, and solvability are sensitive to Levi geometry. In favorable cases, analytic estimates imply that CR functions extend to holomorphic functions or that the manifold can be realized as a real submanifold of complex space.

CR geometry studies which structures are equivalent under diffeomorphisms preserving the complex distribution, not merely under coordinate changes in an ambient complex manifold. A pseudo-Hermitian form produces a Tanaka–Webster connection, whose torsion and curvature provide CR analogues of pseudo-Riemannian invariants; their formulas depend on the contact form, while suitable combinations have conformal or invariant meaning.

In three real dimensions, Cartan’s equivalence method yields a fundamental curvature obstruction to local equivalence with the spherical model. In higher dimensions, parabolic geometry and the Fefferman construction connect CR structures to conformal geometry in one higher real dimension. Fefferman’s metric also relates the boundary geometry of a strictly pseudoconvex domain to its complex Monge–Ampère equation and Bergman kernel.¹ These links make CR invariants relevant to both local classification and global analysis.

CR embeddability is a global property and is not automatic, even when the local distribution satisfies the formal integrability condition. Compact CR manifolds may possess few global CR functions, and the obstruction can be expressed through analytic cohomology or failure of suitable estimates.² This separates CR geometry from the study of boundaries that are already presented inside a complex manifold.

A less familiar feature is the dependence of many constructions on the choice of contact form: changing it produces a CR analogue of conformal rescaling and leads to CR versions of Yamabe-type problems. The subject also includes partially integrable and Levi-degenerate structures, where standard positivity arguments fail and finer invariants are required. Its toolkit therefore ranges from Cartan’s moving frames and contact geometry to microlocal analysis, representation theory, and several complex variables.

CR manifold

A real manifold equipped with a formally integrable complex distribution in its complexified tangent bundle.

CR dimension

The complex rank of the distinguished subbundle T^{1,0}.

Levi form

A Hermitian form on the CR distribution derived from commutators or the exterior derivative of a contact form.

Pseudoconvexity

A positivity condition on the Levi form that supports analytic estimates and boundary extension results.

Pseudo-Hermitian structure

A CR structure together with a selected contact form, yielding additional connection and curvature data.