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

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 $TM$ 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.