Quaternionic CR Geometry

TL;DR

This paper introduces a new quaternionic analogue of CR structures, explores their properties, and constructs a canonical connection under certain convexity conditions, extending complex CR geometry concepts to quaternionic settings.

Contribution

It defines quaternionic CR structures, introduces strong pseudoconvexity and pseudohermitian structures, and constructs a canonical connection analogous to the Tanaka-Webster connection.

Findings

Defined quaternionic CR structures and their properties

Established the notion of strong pseudoconvexity and pseudohermitian structures

Constructed a canonical connection under ultra-pseudoconvexity

Abstract

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which is stronger than the strong pseudoconvexity. Comparison to Biquard's quaternionic contact structure is also made.