Transportation on spheres via an entropy formula

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