Conformal quaternionic contact curvature and the local sphere theorem

TL;DR

This paper introduces a new tensor invariant called quaternionic contact conformal curvature, which characterizes when a quaternionic contact manifold is locally equivalent to a standard flat or 3-sasakian structure, extending classical conformal invariants.

Contribution

The paper defines the quaternionic contact conformal curvature tensor and proves it vanishes precisely when the manifold is locally conformally equivalent to the standard model.

Findings

Quaternionic contact conformal curvature tensor is introduced.

Vanishing of this tensor characterizes local conformal equivalence to the standard structure.

Establishes an analogue of the sphere theorem in quaternionic contact geometry.

Abstract

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