Lie groups with flat Gauduchon connections

• DOI: 10.1007/s00209-019-02232-w
• Publication Date: 2019-01-18T21:57:32Z
• Authors: Luigi Vezzoni; Bo Yang; Fangyang Zheng
• Keywords: Mathematics - Differential Geometry; Lie groups; Hermitian manifolds; left-invariant metrics; Lie groups; 01 natural sciences; Left-invariant metrics; Differential Geometry (math.DG); Hermitian manifolds; 0103 physical sciences; FOS: Mathematics; 0101 mathematics

We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in \mathbb{R}$ and $\nabla^c$ and $\nabla^b$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected $2n$-dimensional Lie group $G$ equipped with a left-invariant complex structure $J$ and a left-invariant compatible metric $g$ and we assume that its connection $\nabla^s$ is flat. Our main result states that if either $n$=2 or there exits a $\nabla^s$-parallel left invariant frame on $G$, then $g$ must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $\nabla^s$-flat Hermitian metrics.<br/>10 pages, In this new version, we add Cor 1.6 in the introduction and also an appendix on Kahler flat Lie groups<br/>