Conformal paracontact curvature and the local flatness theorem

TL;DR

This paper introduces a new curvature tensor invariant for paracontact manifolds, establishing a local flatness theorem that characterizes when these manifolds are conformally equivalent to hyperbolic models, extending classical CR geometry results.

Contribution

It defines the pc conformal curvature invariant and proves a local flatness theorem for paracontact manifolds, paralleling the Cartan-Chern-Moser theorem in CR geometry.

Findings

Vanishing of pc conformal curvature characterizes local conformal flatness.

Explicit formula for solutions to the sub-ultrahyperbolic Yamabe equation.

Extension of classical CR structure results to paracontact geometry.

Abstract

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan-Chern-Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in the complex vector space is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.