Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem

TL;DR

This paper investigates the quaternionic contact Yamabe problem, characterizing structures with vanishing Biquard torsion, and explicitly describes conformal deformations on the quaternionic Heisenberg group, advancing understanding of quaternionic contact geometry.

Contribution

It provides a partial solution to the quaternionic contact Yamabe problem, characterizes when the Biquard torsion vanishes, and describes conformal deformations explicitly.

Findings

Vanishing Biquard torsion occurs on 3-Sasakian manifolds.

Explicit conformal deformations on the quaternionic Heisenberg group are described.

A new '3-Hamiltonian form' for infinitesimal automorphisms is introduced.

Abstract

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All conformal deformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A '3-Hamiltonian form' of infinitesimal conformal automorphisms of quaternionic contact structures is presented.