English

Recovering Riemannian Geometry from Diffusion

Differential Geometry 2026-01-27 v1 Analysis of PDEs Probability

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.

Keywords

Cite

@article{arxiv.2601.17166,
  title  = {Recovering Riemannian Geometry from Diffusion},
  author = {Amandip Sangha},
  journal= {arXiv preprint arXiv:2601.17166},
  year   = {2026}
}
R2 v1 2026-07-01T09:18:03.122Z