Conformal paraquaternionic contact curvature and the local flatness theorem

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.