On the integrability of the isotropic almost complex structures and harmonic unit vector fields

TL;DR

This paper investigates the conditions under which isotropic almost complex structures on tangent bundles are integrable and explores harmonic unit vector fields with respect to generalized metrics extending the Sasaki metric.

Contribution

It provides new results on the integrability of isotropic almost complex structures and characterizes harmonic unit vector fields for a class of generalized metrics.

Findings

Conditions for integrability of isotropic almost complex structures.

Characterization of harmonic unit vector fields under generalized metrics.

Extension of harmonic vector field theory beyond Sasaki metric.

Abstract

Aguilar introduced isotropic almost complex structures $J_{\delta , \sigma}$ on the tangent bundle of a Riemannian manifold $(M, g)$. In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics $g_{\delta , \sigma}$ on $T M$ which are a generalization of the Sasaki metric. Moreover, the notion of a harmonic unit vector field is introduced with respect to these metrics like as the Sasaki metric and the necessary and sufficient conditions for a unit vector field to be a harmonic unit vector field are obtained.