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Recovering Riemannian Geometry from Diffusion

Amandip Sangha

TL;DR

The paper addresses how to recover full Riemannian geometry from diffusion data without pre-assigned metrics. It develops an intrinsicBakry–Émery $\Gamma$-calculus-based procedure in which the first-order carré du champ $\Gamma$ determines a unique metric $g$, while the second-order operator $\Gamma_2$ encodes curvature and, together with $\nabla$, fixes the Levi-Civita connection; symmetry of the diffusion then determines the reference measure as $\mu=\rho\,\mathrm{vol}_g$. The diffusion semigroup $$(P_t)_{t\ge0}$$ thus determines the manifold $(M,g,\mu)$ up to a global isometry, yielding an information-theoretic invariant of Riemannian geometry. This framework reframes geometry as the structure organizing information flow under diffusion and lays groundwork for extending reconstruction to broader diffusion settings and non-smooth analogues.

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.

Recovering Riemannian Geometry from Diffusion

TL;DR

The paper addresses how to recover full Riemannian geometry from diffusion data without pre-assigned metrics. It develops an intrinsicBakry–Émery -calculus-based procedure in which the first-order carré du champ determines a unique metric , while the second-order operator encodes curvature and, together with , fixes the Levi-Civita connection; symmetry of the diffusion then determines the reference measure as . The diffusion semigroup thus determines the manifold up to a global isometry, yielding an information-theoretic invariant of Riemannian geometry. This framework reframes geometry as the structure organizing information flow under diffusion and lays groundwork for extending reconstruction to broader diffusion settings and non-smooth analogues.

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.
Paper Structure (18 sections, 8 theorems, 153 equations)

This paper contains 18 sections, 8 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diffusion semigroup
  4. Information-theoretic quantities
  5. The noisy-label model
  6. Mutual information under diffusion flow
  7. Information dissipation identity
  8. Smooth structure from information
  9. Recovering the Riemannian Metric
  10. The intrinsic distance associated with diffusion
  11. Recovering the Riemannian Structure from Information Diffusion
  12. Recovering the metric tensor
  13. Curvature information from $\Gamma_2$
  14. Recovering the Levi--Civita connection
  15. Recovery of the reference measure
  16. ...and 3 more sections

Key Result

Lemma 2.4

Assume the one-variable chain rule in Assumption 2.2, together with bilinearity and the Leibniz rule. Then for every smooth $\Phi:\mathbb{R}^m\to\mathbb{R}$ and every $F=(f_1,\dots,f_m)\in C^\infty(M;\mathbb{R}^m)$,

Theorems & Definitions (23)

  • Remark 2.3
  • Lemma 2.4: Multivariate chain rule
  • proof
  • Proposition 2.5: Equivalence with generator formula
  • proof
  • Theorem 4.1: Metric recovered from $\Gamma$
  • proof
  • Remark 4.2
  • Remark 4.3: Generator representation of $\Gamma_2$
  • Theorem 4.4: Levi--Civita connection recovered
  • ...and 13 more