Harmonic Hermitian structures on Riemannian manifolds with skew-torsion

Abstract

We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space considered with a natural family of Riemannian metrics defined by means of the metric and the given connection on the base manifold.

arXiv commentan example added; to appear in Mediterr. J. Math. arXiv admin note: text overlap with arXiv:1504.01610