The Large-$N$ Limit of Ricci curvature in the Zeitlin Model on the Torus

Abstract

The Euler equations of hydrodynamics describe the motion of an ideal incompressible fluid. Arnold observed that their solutions are geodesics on the group of volume-preserving diffeomorphisms of the fluid domain, equipped with the $L^2$ metric, so that the curvature of this group governs how nearby fluid motions separate. Sectional curvature takes both signs, and Ricci curvature, its average over all directions, is the natural measure of which sign prevails, but in infinite dimensions the sum defining it diverges. When the fluid domain is the flat torus, we define Ricci curvature through Zeitlin's finite-dimensional models, which approximate the Euler equations by geodesic equations on the matrix groups $SU(N)$. We compute the Ricci curvature of these models exactly and prove that, normalized by dimension, it converges as $N \rightarrow \infty$ to an explicit integral. The limit is strictly negative in every Fourier mode and decreases logarithmically with the wavenumber, the behavior conjectured in \cite{lichtenfelz2026ricci} for the sphere. We also show that the diameter of $SU(N)$ in the Zeitlin metric grows at most linearly in $N$.