A note on compact homogeneous manifolds with Bismut parallel torsion.

In this article, we investigate the class of Hermitian manifolds whose Bismut connection has parallel torsion (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 BTP manifolds. Moreover we show that certain compact flag manifolds are BTP if and only if the metric is Kahler or induced by the Cartan-Killing form and we then characterize 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.