The Riemannian geometry of the probability space of the unit circle
André Magalhães de Sá Gomes, Christian S. Rodrigues, Luiz A. B. San Martin
TL;DR
The paper develops a complete intrinsic Riemannian picture for the Wasserstein space $P(S^1)$ by leveraging the Peter–Weyl decomposition to work with a Fourier-based basis, enabling explicit expressions for the metric, Lie brackets, Levi-Civita connection, and Christoffel symbols. It demonstrates that geodesics in $P(S^1)$ satisfy a Hamilton–Jacobi type equation and that density evolution reduces to an inviscid Burgers equation in the natural fluid-interpretation variables, with periodicity constraining solutions. The main results include explicit matrix coefficients of the Otto metric, the Levi-Civita connection in terms of Fourier modes, detailed Christoffel-symbol formulas, the geodesic equations, and the remarkable finding that $P(S^1)$ is flat ($\overline{R}=0$). This work deepens the bridge between optimal transport geometry and infinite-dimensional differential geometry, with potential implications for dynamical systems on circular domains and related Lie-group settings.
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.