Harmonic and minimal vector fields on tangent and unit tangent bundles

We show that the geodesic flow vector field on the unit tangent sphere bundle of a two-point homogeneous space is both minimal and harmonic and determines a harmonic map. For a complex space form, we exhibit additional unit vector fields on the unit tangent sphere bundle with those properties. We find the same results for the corresponding unit vector fields on the pointed tangent bundle. Moreover, the unit normal to the sphere bundles in the pointed tangent bundle of any Riemannian manifold always enjoys those properties.