Minimality, harmonicity and CR geometry for Reeb vector fields

Let (M, g) be a Riemannian manifold and T_1M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied considering on T_1M the Sasaki metric G_S. This metric, and other well known Riemannian metrics on T_1M, are particular examples of Riemannian natural metrics. In this paper we equip T_1M with a Riemannian natural metric G and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kaehler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any G belongs to a family depending on two parameters of metrics of Kaluza-Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map ξ : (M, g) → (T_1M, G) for any Riemannian natural metric G. Besides, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudohermitian map; if in addition ξ is geodesic then ξ : (M, g) → (T_1M, G) is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and G is a special metric of Kaluza-Klein type. Finally, in the last Section, we get that there is a family of strictly pseudoconvex CR structure on T_1S^{2n+1} depending on two-parameter for which a Hopf vector field ξ determines a pseudoharmonic map (in the sense of Barletta-Dragomir-Urakawa) from S^{2n+1} to T_1S^{2n+1}.