*Accepted Paper*

**Inserted:** 17 mar 2020

**Last Updated:** 16 jul 2020

**Journal:** Math. Z.

**Year:** 2020

**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 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 first Chern class vanishes, the pluriclosed flow preserves the Vaisman condition on complex surfaces.