Harmonicity of the complex structure on product of trans-Sasakian manifolds
Nidhi Yadav, Punam Gupta, R. K. Gangele
TL;DR
The paper studies the harmonicity of the complex structure on the product of two trans-Sasakian manifolds within the transverse geometry framework. It constructs the Hermitian structure $(J_{a,b},g_{a,b})$ on $M_1\times M_2$ via Morimoto–Tsukada-type construction and derives the explicit Levi-Civita connection in terms of the factors' connections. The harmonicity condition is formulated as $[J,\nabla^*\nabla J]=0$, with a computable codifferential $\delta J$ and the result $\nabla_{\delta J} J=0$, yielding concrete criteria in terms of the trans-Sasakian parameters $(\alpha_i,\beta_i)$ and vertical/horizontal directions. The authors show that many product cases are astheno-Kähler and admit a harmonic complex structure, highlighting the structural interplay between trans-Sasakian types and harmonic Hermitian geometry.
Abstract
In this paper, we investigate the transverse geometry of trans-Sasakian manifolds and present several significant findings. We analyze the Levi-Civita connection associated with the metric on the product manifold of two trans-Sasakian manifolds. We outline the conditions under which the complex structure is harmonic on the product manifold. Notably, we also offer valuable insights into the harmonicity of the complex structure within the context of astheno- Kahler manifolds.