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\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^s =(1-\frac{s}{2})\nabla ^c + \frac{s}{2}\nabla ^b$$\end{document} is flat, where s∈R\documentclass[12pt]{minimal}
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\begin{document}$$s\in {\mathbb {R}}$$\end{document} and ∇c\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^c$$\end{document} and ∇b\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^b$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^s$$\end{document} is flat. Our main result states that if either n=2 or there exits a ∇s\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^s$$\end{document}-parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ∇s\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^s$$\end{document}-flat Hermitian metrics.
