Quaternionic contact manifolds with a closed fundamental 4-form

TL;DR

This paper establishes a precise condition under which quaternionic contact manifolds have a closed fundamental 4-form, linking it to the vanishing of torsion and characterizing certain geometric structures.

Contribution

It proves that the fundamental 4-form is closed if and only if the torsion endomorphism vanishes, characterizing locally qc homothetic to 3-Sasakian structures in high dimensions.

Findings

Fundamental 4-form is closed iff torsion endomorphism vanishes.

Characterization of quaternionic contact structures related to 3-Sasakian structures.

Applicable for manifolds of dimension at least eleven.

Abstract

We show that the fundamental 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion endomorphism of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3-Sasakian structures.