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An Information-Theoretic Reconstruction of Curvature

Amandip Sangha

TL;DR

This work develops an intrinsic information-theoretic method to recover the full Riemann curvature tensor from the small-time diffusion of heat. By restricting heat flow from a point to a two-dimensional plane via the exponential map and comparing with the Euclidean model through relative entropy, the authors obtain a linear small-time expansion D_σ(t) = $\frac{1}{6}$ $\operatorname{Scal}(x)$ $t$ + $C$ $K(x,σ)$ $t$ + O($t^{3/2}$) with $C = -\frac{2}{3}$, and assemble these directional coefficients into the curvature-information tensor I_x, which exactly matches the classical curvature operator R_x on Λ^2 T_xM. This provides an analytic, Jacobi-field-free view in which curvature arises as an infinitesimal information defect of heat flow, and yields explicit averaging formulas to recover scalar and Ricci curvatures from diffusion data. The framework bridges geometric analysis and information theory and suggests diffusion-based approaches to curvature reconstruction and related geometric inequalities.

Abstract

We develop an intrinsic information-theoretic approach for recovering Riemannian curvature from the small-time behaviour of heat diffusion. Given a point and a two-plane in the tangent space, we compare the heat mass transported along that plane with its Euclidean counterpart using the relative entropy of finite measures. We show that the leading small-time distortion of this directional entropy encodes precisely the local curvature of the manifold. In particular, the planar information imbalance determines both the scalar curvature and the sectional curvature at a point, and assembling these directional values produces a bilinear tensor that coincides exactly with the classical Riemannian curvature operator. The method is entirely analytic and avoids Jacobi fields, curvature identities, or variational formulas. Curvature appears solely through the behaviour of heat flow under the exponential map, providing a new viewpoint in which curvature is realized as an infinitesimal information defect of diffusion. This perspective suggests further connections between geometric analysis and information theory and offers a principled analytic mechanism for detecting and reconstructing curvature using only heat diffusion data.

An Information-Theoretic Reconstruction of Curvature

TL;DR

This work develops an intrinsic information-theoretic method to recover the full Riemann curvature tensor from the small-time diffusion of heat. By restricting heat flow from a point to a two-dimensional plane via the exponential map and comparing with the Euclidean model through relative entropy, the authors obtain a linear small-time expansion D_σ(t) = + + O() with , and assemble these directional coefficients into the curvature-information tensor I_x, which exactly matches the classical curvature operator R_x on Λ^2 T_xM. This provides an analytic, Jacobi-field-free view in which curvature arises as an infinitesimal information defect of heat flow, and yields explicit averaging formulas to recover scalar and Ricci curvatures from diffusion data. The framework bridges geometric analysis and information theory and suggests diffusion-based approaches to curvature reconstruction and related geometric inequalities.

Abstract

We develop an intrinsic information-theoretic approach for recovering Riemannian curvature from the small-time behaviour of heat diffusion. Given a point and a two-plane in the tangent space, we compare the heat mass transported along that plane with its Euclidean counterpart using the relative entropy of finite measures. We show that the leading small-time distortion of this directional entropy encodes precisely the local curvature of the manifold. In particular, the planar information imbalance determines both the scalar curvature and the sectional curvature at a point, and assembling these directional values produces a bilinear tensor that coincides exactly with the classical Riemannian curvature operator. The method is entirely analytic and avoids Jacobi fields, curvature identities, or variational formulas. Curvature appears solely through the behaviour of heat flow under the exponential map, providing a new viewpoint in which curvature is realized as an infinitesimal information defect of diffusion. This perspective suggests further connections between geometric analysis and information theory and offers a principled analytic mechanism for detecting and reconstructing curvature using only heat diffusion data.

Paper Structure

This paper contains 15 sections, 6 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian geometry and sectional curvature
  4. Heat kernel and small-time asymptotics
  5. Relative entropy for heat mass distributions
  6. The curvature–-information functional
  7. Heat mass measure on a two–-plane
  8. Normal coordinates and metric expansion
  9. Heat kernel parametrix in normal coordinates.
  10. Expansion of the heat mass on a plane.
  11. Information tensor and reconstruction of the curvature operator
  12. Structural properties of the curvature--information tensor
  13. Algebraic structure.
  14. Ricci and scalar curvature as averaged information loss.
  15. Concluding remarks

Key Result

Theorem 3.2

For every $x\in M$ and every oriented two--plane $\sigma\subset T_xM$, the curvature--information functional admits the small--time expansion with the universal constant $C=-\tfrac{2}{3}$. The scalar--curvature term is independent of $\sigma$, while the anisotropic component encodes the sectional curvature of $\sigma$.

Theorems & Definitions (10)

  • Definition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2: Ricci curvature from directional information loss
  • Proposition 5.3: Scalar curvature as averaged information loss