论文标题
关于满足Kähler的条件的Hermitian流形的一些评论
Some remarks on Hermitian manifolds satisfying Kähler-like conditions
论文作者
论文摘要
我们研究了Hermitian指标，其Bismut连接$ \ nabla^b $满足了与SKT条件和Bimut连接扭转的平行性有关的第一个Bianchi身份。我们获得了复杂表面的表征，该表面承认遗传学指标，其双重连接满足了第一个bianchi的身份和条件$ r^b（x，x，y，y，z，w）= r^b（jx，jy，jy，z，z，w）$，对于每一个切线vectors $ x，y，y，z，w $最近在[D. Angella，A。Otal，L。Ugarte，R。Villacampa，在Gauduchon与Kähler样曲率的连接上，出现在Commun中。肛门。 Geom。，Arxiv：1809.02632 [Math.dg]]，[Q. Zhao，F。Zheng，Strominger Connection和Pluriclosed Metrics，Arxiv：1904.06604 [Math.dg]]，[S。 T. Yau，Q。Zhao，F。Zheng，在StromingerKähler的歧管上，带有堕落的扭转，Arxiv：1908.05322 [Math.dg]]。在[R. M. Arroyo，R。Lafuente，均质多裂流的长期行为，Proc。伦敦数学。 Soc。 （3），119，（2019年），266-289]，我们构建了满足BismutKähler状态的Hermitian流形的新例子。此外，我们证明了与复杂表面和几乎阿贝尔谎言组上的多瓣流有关的一些结果。特别是，我们表明，如果初始度量具有恒定的标态曲率，则多瓣流将Vaisman条件保留在复杂表面上。
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 [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with Kähler-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 Kähler-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 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.
