Lie Groups with flat Gauduchon connections

Luigi Vezzoni, Bo Yang, Fangyang Zheng

Published 2018-05-12Version 1

In \cite{YZ-Gflat} it is considered the question of classifying compact Hermitian manifolds with flat $s$-Gauduchon connection, which is $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$, where $\nabla^c$ and $\nabla^b$ are the Chern and Bismut connections respectively. In this note, we consider even-dimensional connected Lie group $G$ equipped with a left invariant metric $g$ and a compatible left invariant complex structure $J$. We show that when $s\neq 0, 2$ and when there is a left invariant $\nabla^s$-parallel frame, the manifold must be the flat K\"ahler space.