Recovering Riemannian Geometry from Diffusion

TL;DR

This paper demonstrates how the intrinsic properties of a diffusion semigroup can be used to reconstruct the full Riemannian geometry of a manifold, revealing deep geometric insights from diffusion data alone.

Contribution

It provides a method to recover the entire Riemannian structure solely from diffusion operators and their calculus, without prior metric assumptions.

Findings

Carre du champ determines a unique Riemannian metric

Iterated carre du champ encodes curvature information

Diffusion symmetry fixes Levi-Civita connection and measure

Abstract

We present an intrinsic reconstruction of Riemannian geometry from a symmetric, strongly local diffusion semigroup. Starting from a diffusion operator and its associated first- and second-order diffusion calculus, we recover the full weighted Riemannian structure of the underlying manifold. In particular, we show that the carre du champ determines a unique smooth Riemannian metric, that the iterated carre du champ encodes curvature, and that the symmetry of the diffusion fixes the Levi-Civita connection and reference measure. As a consequence, the diffusion semigroup determines the global Riemannian manifold uniquely up to isometry. The results provide an information-theoretic perspective on differential geometry in which geometric structure emerges from the intrinsic behavior of diffusion, without assuming any prior metric or coordinate description.