Curvature characterization of Hermitian manifolds with Bismut parallel torsion

TL;DR

This paper characterizes Hermitian manifolds with Bismut connection having parallel torsion, providing a curvature-based criterion, examples, properties, and a classification of certain threefolds.

Contribution

It offers a necessary and sufficient condition for BTP manifolds using Bismut curvature, along with examples and classification results.

Findings

Derived a curvature-based criterion for BTP manifolds.

Presented examples and discussed properties of BTP manifolds.

Classified non-balanced BTP threefolds.

Abstract

In this article, we study Hermitian manifolds whose Bismut connection has parallel torsion, which will be called {\em Bismut torsion parallel manifolds,} or {\em 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).