On Bismut Flat Manifolds

TL;DR

This paper classifies all compact Hermitian manifolds with flat Bismut connection, revealing their structure as Lie groups with bi-invariant metrics and identifying specific examples like Hopf surfaces and Calabi-Eckmann threefolds.

Contribution

It provides a complete classification of Bismut flat compact Hermitian manifolds, showing their universal cover is a Lie group with special geometric structures.

Findings

Isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces.

Central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

Torsion tensor must be parallel in such manifolds.

Abstract

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.