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 Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ${\nabla }^s$-flat Hermitian metrics.

10 pages, In this new version, we add Cor 1.6 in the introduction and also an appendix on Kahler flat Lie groups

References in corpus (3)

• Invariant Solutions to the Strominger System on Complex Lie Groups and Their Quotients
• On Curvature Tensors of Hermitian Manifolds
• On compact Hermitian manifolds with flat Gauduchon connections