Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds

TL;DR

This paper characterizes compact harmonic RCD(K,N) spaces, showing they are smooth Riemannian manifolds under specific harmonicity conditions related to heat kernel dependence and volume intersection properties.

Contribution

It establishes that certain harmonic RCD(K,N) spaces are actually smooth Riemannian manifolds, extending classical results to a non-smooth setting.

Findings

Harmonic RCD(K,N) spaces satisfying specific conditions are smooth manifolds.

Heat kernel dependence on distance and time characterizes harmonicity.

Volume intersection properties imply manifold structure.

Abstract

In this paper, we study harmonic RCD$(K,N)$ spaces as the counterpart of harmonic Riemannian manifolds with Ricci curvature bounded from below. We prove that a compact RCD$(K,N)$ space is isometric to a smooth closed Riemannian manifold if it satisfies either of the following harmonicity conditions:(1) the heat kernel $\rho(x,y,t)$ depends only on the variable $t$ and the distance between points $x$ and $y$; (2) the volume of the intersection of two geodesic balls depends only on their radii and the distance between their centers.