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Lie groups with flat Gauduchon connections

  • Luigi Vezzoni
  • Bo YangEmail author
  • Fangyang Zheng
Article
  • 18 Downloads

Abstract

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.

Keywords

Hermitian manifolds Lie groups left-invariant metrics 

Mathematics Subject Classification

Primary 53C55 

Notes

Acknowledgements

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

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

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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