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In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, andVillacampa recently, which states that if the Strominger connection (also knownas Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in thesense that its curvature tensor obeys all the symmetries of the curvature of aK\"ahler manifold, then the metric must be pluriclosed. What we actually showedis a bit more: for any given Hermitian manifold, the Strominger K\"ahler-likecondition is equivalent to the pluriclosedness of the metric plus theparallelness of the torsion.

Strominger connection and pluriclosed metrics