In this paper, we study a special type of compact Hermitian manifolds that are Strominger Ka.hler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Ka.hler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Ka.hler manifold. Previously, we have shown that any SKL manifold (Mn, g) is always pluriclosed, and when the manifold is compact and g is not Ka.hler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent t...
