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<p>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 Kähler or induced by the Cartan-Killing form and we then characterize BTP invariant metrics on compact semisimple Lie groups which are Hermitian with respect to 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.</p>

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A note on compact homogeneous manifolds with Bismut parallel torsion

  • 期刊论文

  • 发表日期：

    2025年2月25日

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  • Fabio Podestà,
  • Fangyang Zheng

摘要

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 Kähler or induced by the Cartan-Killing form and we then characterize BTP invariant metrics on compact semisimple Lie groups which are Hermitian with respect to 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.