Lie groups with flat Gauduchon connections

We pursue the research line proposed in Yang and Zheng (Acta. Math. Sinica (English Series), 34(8):1259–1268, 2018) about the classification of Hermitian manifolds whose s-Gauduchon connection ∇s=(1-s2)∇c+s2∇b is flat, where s∈ R and ∇ c and ∇ 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 ∇ s is flat. Our main result states that if either n=2 or there exits a ∇ s-parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ∇ s-flat Hermitian metrics.