We develop a detailed and rigorous theory of the information Ricci flow on the infinite-dimensional manifold of smooth positive probability densities on a compact Riemannian manifold. We construct the Fisher--Rao metric, derive the Levi--Civita connection and the geodesic equation by explicit variational calculus, and prove local well-posedness in Sobolev spaces. We then introduce an information Ricci flow coupling the base metric and the density, derive it as a gradient flow of an information entropy functional of Perelman type, and prove a monotonicity formula by a complete computation. We define information singularities and information horizons and establish an information area law. Several original equations are introduced and derived in full detail, providing a mathematically solid foundation for the information-as-matter paradigm.
Y. Li (Mon,) studied this question.