Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds

Adrián Andrada, Alejandro Tolcachier

Published 2023-01-23Version 1

It is well-known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures $(J_{a,b},g_{a,b})$. We show in this article that the complex structure $J_{a,b}$ is harmonic with respect to $g_{a,b}$, i.e. it is a critical point of the energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally K\"ahler, balanced, strong K\"ahler with torsion, Gauduchon or $k$-Gauduchon ($k\geq 2$). Finally, we provide an expression for the Bismut connection associated to $(J_{a,b}, g_{a,b})$ in terms of the characteristic connections on each Sasakian factor.