$H$-contact unit tangent sphere bundles of Riemannian manifolds
  A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein. 
Yuri Nikolayevsky, Jeong Hyeong Park
math.DG