On compact Hermitian manifolds with flat Gauduchon connections

Abstract

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call ${\nabla }^s=(1-\frac{s}{2}){\nabla }^c+\frac{s}{2}{\nabla }^b$ the $s$-Gauduchon connection of $M$, where ${\nabla }^c$ and ${\nabla }^b$ are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat $s$-Gauduchon connection. This is related to a question asked by Yau \cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are classified respectively by Boothby \cite{Boothby} in the 1950s or by Q. Wang and the authors recently \cite{WYZ}. In this article, we observe that if either $s\ge 4+2\sqrt{3}\approx 7.46$ or $s\le 4-2\sqrt{3}\approx 0.54$ and $s\ne 0$, then $g$ is Kähler. We also show that, when $n=2$, $g$ is always Kähler unless $s=2$. Note that non-Kähler compact Bismut flat surfaces are exactly those isosceles Hopf surfaces by \cite{WYZ}.

9 pages. This preprint was submitted to Acta Mathematica Sinica, a special issue dedicated to Professor Qikeng Lu

References in corpus (2)

• On Curvature Tensors of Hermitian Manifolds
• Minimal complex surfaces with Levi-Civita Ricci-flat metrics

Cited by in corpus (1)

• Lie Groups with flat Gauduchon connections