On Bismut--Ambrose--Singer manifolds

TL;DR

This paper classifies and constructs Bismut--Ambrose--Singer manifolds, a special class of Hermitian manifolds with parallel torsion and curvature, across various homogeneous and pluriclosed cases.

Contribution

It provides a canonical reduction theorem and comprehensive classification results for complete, simply-connected BAS manifolds in multiple geometric settings.

Findings

Classified simply-connected BAS manifolds in compact, semisimple, and nilpotent cases.

Constructed new BAS manifolds combining different geometric structures.

Classified complete, simply-connected, pluriclosed BAS manifolds.

Abstract

We investigate Bismut--Ambrose--Singer (BAS) manifolds, namely Hermitian manifolds whose Bismut connection has parallel torsion and parallel curvature. We first establish a canonical reduction theorem for complete, simply-connected BAS manifolds. We then classify simply-connected BAS manifolds in the three fundamental homogeneous settings: the compact case, the non-compact semisimple case, and the nilpotent case. Building on this, we construct BAS manifolds in which these three geometries are combined, generalizing all previously known examples. Finally we classify complete, simply-connected, pluriclosed BAS manifolds.