Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).