Lie Groups with flat Gauduchon connections

Abstract

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 K\"ahler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $\nabla^s$-flat Hermitian metrics.