Harmonicity of unit vector fields with respect to a class of Riemannian metrics

TL;DR

This paper investigates the harmonicity of unit vector fields on Riemannian manifolds equipped with a class of generalized Sasakian metrics induced by isotropic almost complex structures, including explicit calculations and examples.

Contribution

It computes the Levi-Civita connection for these metrics and analyzes the harmonicity conditions of unit vector fields, providing new insights and an illustrative example.

Findings

Derived the Levi-Civita connection for $g_{ ext{delta}, ext{sigma}}$ metrics.

Established conditions for harmonicity of unit vector fields.

Presented a key example satisfying the main theorem.

Abstract

The isotropic almost complex structures induce a Riemannian metric $g_{\delta,\sigma}$ on TM, which are the generalized type of Sasakian metric. In this paper, the Levi-Civita connection of $g_{\delta,\sigma}$ is calculated and the harmonicity of unit vector fields from $(M, g)$ to $(S(M), i*g_{\delta,0})$ is investigated, where $i*g_{\delta,0}$ is a particular type of induced metric $i*g_{\delta,\sigma}$. Finally, an important example is presented which satisfies in main theorem of the paper.