Conformal paraquaternionic contact curvature and the local flatness theorem

TL;DR

This paper introduces a new tensor invariant called paraquaternionic contact conformal curvature, which characterizes local flatness of paraquaternionic contact manifolds, similar to classical curvature tensors in various geometries.

Contribution

It defines the paraquaternionic contact conformal curvature tensor and proves that its vanishing characterizes local conformal flatness of the manifold.

Findings

The tensor is analogous to Weyl and Chern-Moser tensors.

Vanishing of the tensor implies local conformal equivalence to flat models.

Provides a criterion for local flatness in paraquaternionic contact geometry.

Abstract

A tensor invariant is defined on a paraquaternionic contact manifold in terms of the curvature and torsion of the canonical paraquaternionic connection involving derivatives up to third order of the contact form. This tensor, called paraquaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry, the Chern-Moser tensor in CR geometry, the para contact curvature in para CR geometry and to the quaternionic contact conformal curvature in quaternionic contact geometry. It is shown that a paraquaternionic contact manifold is locally paraquaternionic contact conformal to the standard flat paraquaternionic contact structure on the paraquaternionic Heisenberg group, or equivalently, to the standard para 3-Sasakian structure on the paraquaternionic pseudo-sphere iff the paraquaternionic contact conformal curvature vanishes.