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 backward difference s=(1-s2) backward difference c+s2 backward difference b is flat, where s is an element of R and backward difference c and backward difference 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 backward difference s is flat. Our main result states that if either n=2 or there exits a backward difference s-parallel left invariant frame on G, then g must be Kahler. This result demonstrates rigidity properties of some complete Hermitian manifolds with backward difference s-flat Hermitian metrics.