Geodesics and killing vector fields on the tangent sphere bundle

We show that any Killing vector field on the unit tangent sphere bundle with Sasaki metric of a space of constant curvature kappa not equal 1 is fiber preserving by studying some property of geodesics on the bundle. As a consequence, any Killing vector field on the unit tangent sphere bundle of a space of constant curvature kappa not equal 1 can be extended to a Killing vector field on the tangent bundle.