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

TL;DR

This paper investigates harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds, characterizing their geometric properties and conditions for various special structures, and providing new examples of Calabi-Yau with torsion manifolds.

Contribution

It demonstrates that the complex structures on these products are harmonic and characterizes when they are conformally Kähler, balanced, or have special torsion properties, also analyzing the Bismut connection.

Findings

Complex structures are harmonic with respect to the Hermitian metrics.

Conditions for Hermitian structures to be conformally Kähler, balanced, or have torsion.

Vanishing of Bismut-Ricci tensors when factors are η-Einstein, leading to new Calabi-Yau with torsion examples.

Abstract

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 Dirichlet 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 study the Bismut connection associated to $(J_{a,b}, g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor $\operatorname{Ric}^B$ and the Bismut-Ricci form $\rho^B$. We show that these tensors vanish if and only if each Sasakian factor is $\eta$-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.