  The paper proves transportation inequalities for probability measures on
spheres for the Wasserstein metrics with respect to cost functions that are
powers of the geodesic distance. Let $\mu$ be a probability measure on the
sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}dx$ where $dx$ is the rotation
invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {\kappa_U}I$,
where $\kappa_U>0$. Then any probability measure $\nu$ of finite relative
entropy with respect to $\mu$ satisfies ${\hbox{Ent}}(\nu\mid\mu) \geq
(\kappa_U/2)W_2(\nu, \mu )^2$. The proof uses an explicit formula for the
relative entropy which is also valid on connected and compact $C^\infty$ smooth
Riemannian manifolds without boundary. A variation of this entropy formula
gives the Lichn\'erowicz integral.
