The Riemannian geometry of the probability space of the unit circle

Abstract

This paper explores the Riemannian geometry of the Wasserstein space of the circle, namely $P(S^{1})$, the set of probability measures on the unit circle endowed with the 2-Wasserstein metric. Building on the foundational work of Otto, Lott, and Villani, the authors developed in another work an intrinsic framework for studying the differential geometry of Wasserstein spaces of compact Lie groups, making use of the Peter-Weyl Theorem. This formalism allowed them to explicit an example in this paper. Key contributions include explicit computations of the Riemannian metric matrix coefficients, Lie brackets, and the Levi-Civita connection, along with its associated Christoffel symbols. The geodesic equations and curves with constant velocity fields are analysed, expliciting their PDEs. Notably, the paper demonstrates that $P(S^{1})$ is flat, with vanishing curvature. These results provide a comprehensive geometric understanding of $P(S^{1})$, connecting optimal transport theory and differential geometry, with potential applications in dynamical systems.