On Bismut Flat Manifolds
Differential Geometry
2023-03-31 v3
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.
Keywords
Cite
@article{arxiv.1603.07058,
title = {On Bismut Flat Manifolds},
author = {Qingsong Wang and Bo Yang and Fangyang Zheng},
journal= {arXiv preprint arXiv:1603.07058},
year = {2023}
}
Comments
In this 3rd version, we add a lemma on Hermitian surfaces with flat Riemannian connection. References are updated and typos corrected