Harmonic Complex Structures and Special Hermitian Metrics on Products of Sasakian Manifolds

It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures \((J_{a,b},g_{a,b})\) $(J_{a,b},g_{a,b})$ . We show in this article that the complex structure \(J_{a,b}\) $J_{a,b}$ is harmonic with respect to \(g_{a,b}\) $g_{a,b}$ , i.e., it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon ( \(k\ge 2\) $k\ge 2$ ). Finally, we study the Bismut connection associated to \((J_{a,b}, g_{a,b})\) $(J_{a,b},g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor \({\text {Ric}}^B\) ${\text{Ric}}^B$ and the Bismut-Ricci form \(\rho ^B\) ${\rho }^B$ . We show that these tensors vanish if and only if each Sasakian factor is \(\eta \) $\eta$ -Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.