Lie groups with flat Gauduchon connections

  • Luigi Vezzoni
  • Bo YangEmail author
  • Fangyang Zheng


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 \(\nabla ^s =(1-\frac{s}{2})\nabla ^c + \frac{s}{2}\nabla ^b\) 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.


Hermitian manifolds Lie groups left-invariant metrics 

Mathematics Subject Classification

Primary 53C55 



We are grateful to an anonymous referee for providing helpful suggestions to improve the exposition of this paper.


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Copyright information

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Authors and Affiliations

  1. 1.Dipartimento di Matematica G. PeanoUniversitȧ Di TorinoTurinItaly
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina
  3. 3.Department of MathematicsThe Ohio State UniversityColumbusUSA

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