Rigidity of Curvature Bounds of Quotient Spaces Of Isometric Actions

Let $G\curvearrowright M$ be an isometric action of a Lie Group on a complete orientable Riemannian manifold. We disintegrate absolutely continuous measures with respect to the volume measure of $M$ along the principal orbits of $G\curvearrowright M$ and define a functional on the probability measures with support on the principal orbits of the action to further prove that the convexity properties of this functional guarantees necessary and sufficient conditions to the Ricci curvature of $M$ to be bound below by a given real number $K$.