  We show, using two different approaches, that there exists a family of
Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics
of constant curvature on its unit tangent bundle. In other words, given such a
metric on the tangent bundle of a two-sphere, the Hopf map is identified with a
Riemannian submersion from the universal covering space of the unit tangent
bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent
bundle of a hyperbolic plane is also presented.
