arXiv 1709.025302017-09-08math.DG

具有平坦Gauduchon联络的紧致Hermitian流形

On compact Hermitian manifolds with flat Gauduchon connections

Bo Yang, Fangyang Zheng

备注：: 9 pages. This preprint was submitted to Acta Mathematica Sinica, a special issue dedicated to Professor Qikeng Lu
发表信息：: Acta. Math. Sinica (English Series), 34 (2018), no.8, 1259-1268

摘要：

给定一个厄米特流形$(M^n,g)$，高杜雄联络是连接陈联络和毕斯穆特联络的单参数厄米特联络族。我们将$\nabla^s = (1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$称为$M$的$s$ -Gauduchon连接，其中$\nabla^c$和$\nabla^b$分别是Chern和Bismut连接。很自然地会问，紧致厄米流形何时可以有平坦的$s$ -Gauduchon联络。这与Yau \cite{Yau}提出的一个问题有关。$s=0$（扁平Chern连接）或$s=2$（扁平Bismut连接）病例分别由Boothby \cite{Boothby}在20世纪50年代或Q. Wang和作者最近完成了\cite{WYZ} .在这篇文章中，我们观察到，如果$s\geq 4+2\sqrt{3} \approx 7.46$或$s\leq 4-2\sqrt{3}\approx 0.54$和$s\neq 0$，那么$g$是Kähler。我们还表明，当$n=2$，$g$总是凯勒，除非$s=2$。注意，非Kähler紧Bismut平坦曲面正是由\cite{WYZ}得到的等腰Hopf曲面。

英文摘要：

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 $s\geq 4+2\sqrt{3} \approx 7.46$ or $s\leq 4-2\sqrt{3}\approx 0.54$ and $s\neq 0$, 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}.

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