Pseudoconformal geometry of hypersurfaces in C.

Area of Science:

• Differential Geometry
• Complex Analysis
• Geometric Analysis

Background:

• The study focuses on the pseudoconformal geometry (CR structure) of real hypersurfaces in complex space C(n+1).
• It builds upon foundational work by Cartan, Tanaka, and Chern regarding connections on principal bundles over these hypersurfaces.

Purpose of the Study:

• To provide an alternative formulation of the Cartan-Tanaka-Chern theorem on normalized Cartan connections.
• To establish a connection between geometric chains and light rays in a related metric space.
• To classify specific types of homogeneous manifolds and investigate deformations of complex structures.

Main Methods:

• Review of pseudoconformal geometry and CR structures.
• Alternative formulation of the Cartan-Tanaka-Chern theorem.
• Analysis of chains derived from Cartan connections.
• Classification of simply connected homogeneous manifolds locally CR equivalent to the sphere.
• Investigation of moduli for complex structure deformations on the unit ball in C(n+1).

Main Results:

• An alternative formulation of the Cartan-Tanaka-Chern theorem is presented.
• Chains are shown to be projections of light rays from a conformal class of Lorentz metrics.
• A classification of simply connected homogeneous manifolds locally CR equivalent to the sphere is provided.
• A theorem concerning moduli for deformations of the complex structure on the ball in C(n+1) is established.

Conclusions:

• The research offers new perspectives on the relationship between pseudoconformal geometry, connections, and related physical concepts like light rays.
• The classification and deformation results contribute to a deeper understanding of the geometry of complex manifolds.
• This work bridges concepts from differential geometry, complex analysis, and potentially theoretical physics.