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Some remarks on Hermitian manifolds satisfying Kähler-like conditions
Mathematische Zeitschrift ( IF 0.8 ) Pub Date : 2020-08-07 , DOI: 10.1007/s00209-020-02598-2
Anna Fino , Nicoletta Tardini

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 K\"ahler-like, have been recently studied in [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K\"ahler-like curvature, to appear in Commun. Anal. Geom., arXiv:1809.02632 [math.DG]], [Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604 [math.DG]], [S. T. Yau, Q. Zhao, F. Zheng, On Strominger K\"ahler-like manifolds with degenerate torsion, arXiv:1908.05322 [math.DG]]. Using the characterization of SKT almost abelian Lie groups in [R. M. Arroyo, R. Lafuente, The long-time behavior of the homogeneous pluriclosed flow, Proc. London Math. Soc. (3), 119, (2019), 266-289], we construct new examples of Hermitian manifolds satisfying the Bismut K\"ahler-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.

中文翻译:

关于满足类 Kähler 条件的 Hermitian 流形的一些评论

我们研究了 Hermitian 度量,其 Bismut 连接 $\nabla^B$ 满足与 SKT 条件相关的第一个 Bianchi 恒等式和 Bimut 连接扭转的并行性。我们获得了允许 Hermitian 度量的复杂表面的表征,其 Bismut 连接满足第一个 Bianchi 恒等式和条件 $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$,对于每个切线向量 $x,y,z,w$,根据 Vaisman 度量。这些条件,也称为 Bismut K\"ahler-like,最近在 [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon 连接与 K\"ahler-like 曲率中进行了研究,出现在社区。肛门。Geom., arXiv:1809.02632 [math.DG]], [Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604 [math.DG]], [ST Yau, Q. Zhao, F. Zheng, On Strominger K\" 具有退化扭转的类 ahler 流形,arXiv:1908.05322 [math.DG]]。使用 [RM Arroyo, R. Lafuente, The long-time behavior of the homogeneous pluriclosed flow, Proc. 伦敦数学。社会。(3), 119, (2019), 266-289],我们构建了满足类 Bismut K\"ahler 条件的 Hermitian 流形的新例子。此外,我们证明了与复杂表面上的多封闭流和几乎是阿贝尔李群。特别是,我们表明,如果初始度量具有恒定的标量曲率,则多闭流在复杂曲面上保持 Vaisman 条件。均质多封闭流的长期行为,Proc。伦敦数学。社会。(3), 119, (2019), 266-289],我们构建了满足类 Bismut K\"ahler 条件的 Hermitian 流形的新例子。此外,我们证明了与复杂表面上的多封闭流和几乎是阿贝尔李群。特别是,我们表明,如果初始度量具有恒定的标量曲率,则多闭流在复杂曲面上保持 Vaisman 条件。均质多封闭流的长期行为,Proc。伦敦数学。社会。(3), 119, (2019), 266-289],我们构建了满足类 Bismut K\"ahler 条件的 Hermitian 流形的新例子。此外,我们证明了与复杂表面上的多封闭流和几乎是阿贝尔李群。特别是,我们表明,如果初始度量具有恒定的标量曲率,则多闭流在复杂曲面上保持 Vaisman 条件。
更新日期:2020-08-07
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