A note on compact homogeneous manifolds with Bismut parallel torsion

TL;DR

This paper studies Hermitian manifolds with Bismut connection having parallel torsion, focusing on homogeneous and compact cases, and characterizes when such manifolds are Kähler or related to Lie group structures.

Contribution

It fully characterizes compact Chern flat BTP manifolds and identifies conditions under which certain flag manifolds and Lie groups exhibit BTP properties.

Findings

Compact Chern flat BTP manifolds are fully characterized.

Certain compact flag manifolds are BTP iff the metric is Kähler or from the Cartan-Killing form.

BTP invariant metrics on compact semisimple Lie groups are characterized.

Abstract

In this article, we investigate the class of Hermitian manifolds whose Bismut connection has parallel torsion ({\rm BTP} for brevity). In particular, we focus on the case where the manifold is (locally) homogeneous with respect to a group of holomorphic isometries and we fully characterize the compact Chern flat {\rm BTP} manifolds. Moreover we show that certain compact flag manifolds are {\rm BTP} if and only if the metric is K\"ahler or induced by the Cartan-Killing form and we then characterize {\rm BTP} invariant metrics on compact semisimple Lie groups which are Hermitian w.r.t. a Samelson structure and are projectable along the Tits fibration. We state a conjecture concerning the question when the Bismut connection of a BTP compact Hermitian locally homogeneous manifold has parallel curvature, giving examples and providing evidence in some special cases.