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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 $d\mu =e^{-U(x)}{\rm d}x$
where ${\rm d}x$
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 Lichnérowicz integral.
