Some remarks on Hermitian manifolds satisfying Kähler-like conditions

We study Hermitian metrics whose Bismut connection ∇ 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 ( J x , J y , 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.