Minimal unit vector fields with respect to Riemannian natural metrics

Abstract

Let (M,g) be a Riemannian manifold. We denote by G˜ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle T1M, 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 T1M equipped with the Sasaki metric G˜S[12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric G˜. In particular, the minimality condition with respect to the Sasaki metric 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 G˜) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to 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 G˜).