The Wasserstein space of the circle is not flat

Abstract

We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.