Curvature characterization of Hermitian manifolds with Bismut parallel torsion

In this article, we study Hermitian manifolds whose Bismut connection has parallel torsion, which will be called { Bismut torsion parallel manifolds,} or { BTP} manifolds for brevity. We obtain a necessary and sufficient condition characterizing BTP manifolds in terms of Bismut curvature tensor alone (Theorem 1.1). We also present examples and discuss some general properties for BTP manifolds, as well as give a classification result for non-balanced BTP threefolds (Theorem 1.16).