Abstract
We study Hermitian metrics whose Bismut connection \(\nabla ^B\) satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition \(R^B(x,y,z,w)=R^B(Jx,Jy,z,w)\), for every tangent vectors x, y, z, w, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in Angella et al. (Commun Anal Geom, to appear, 2018), Yau et al. (2019) and Zhao and Zheng (2019). Using the characterization of SKT almost abelian Lie groups in Arroyo and Lafuente (Proc Lond Math Soc (3) 119:266–289, 2019), we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.
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Acknowledgements
The authors would like to thank Luigi Vezzoni and Mihaela Pilca for useful discussions and suggestions. The authors would like to thank also Quanting Zhao and Fangyang Zheng for useful comments. The authors are grateful to Jeffrey Streets for pointing out an inaccuracy in the first version of Theorem B. The authors are supported by Project PRIN 2017 “Real and complex manifolds: Topology, Geometry and Holomorphic Dynamics”, by project SIR 2014 AnHyC “Analytic aspects in complex and hypercomplex geometry” code RBSI14DYEB, and by GNSAGA of INdAM.
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Fino, A., Tardini, N. Some remarks on Hermitian manifolds satisfying Kähler-like conditions. Math. Z. 298, 49–68 (2021). https://doi.org/10.1007/s00209-020-02598-2
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DOI: https://doi.org/10.1007/s00209-020-02598-2