Transportation on spheres via an entropy formula

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