On homogeneous HKT manifolds and the Einstein condition

TL;DR

This paper characterizes invariant HKT metrics on homogeneous hypercomplex manifolds, proves the existence and uniqueness of an invariant HKT-Einstein metric, and explores conditions for Bismut-parallel torsion and curvature.

Contribution

It provides a complete characterization of invariant HKT metrics on homogeneous hypercomplex manifolds and establishes the existence and uniqueness of an invariant HKT-Einstein metric.

Findings

Existence of a unique invariant HKT-Einstein metric on such manifolds.

Identification of conditions for Bismut-parallel torsion and curvature.

Characterization of invariant strong HKT metrics with these properties.

Abstract

We consider homogeneous hypercomplex manifolds with a transitive action of a compact Lie group and we give a characterization of invariant HKT metrics on them. On every such hypercomplex manifold we prove the existence of an invariant HKT-Einstein metric, which is unique up to scaling. Furthermore, we determine for which invariant HKT metrics the torsion and the curvature of the Bismut connection are Bismut-parallel, showing that invariant strong HKT metrics have this property.