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Strominger connection and pluriclosed metrics
In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and
Villacampa recently, which states that if the Strominger connection (also known
as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the
sense that its curvature tensor obeys all the symmetries of the curvature of a
K\"ahler manifold, then the metric must be pluriclosed. Actually, we show that
Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian
metric plus the parallelness of the torsion, even without the compactness
assumption
Complex nilmanifolds and K\"ahler-like connections
In this note, we analyze the question of when will a complex nilmanifold have
K\"ahler-like Strominger (also known as Bismut), Chern, or Riemannian
connection, in the sense that the curvature of the connection obeys all the
symmetries of that of a K\"ahler metric. We give a classification in the first
two cases and a partial description in the third case. It would be interesting
to understand these questions for all Lie-Hermitian manifolds, namely, Lie
groups equipped with a left invariant complex structure and a compatible left
invariant metric
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