An Information-Theoretic Reconstruction of Curvature

TL;DR

This paper introduces an intrinsic information-theoretic method to recover Riemannian curvature from heat diffusion behavior, linking geometric properties to entropy measures without relying on traditional geometric tools.

Contribution

It presents a novel analytic approach that reconstructs curvature solely from heat flow data, avoiding classical geometric and variational techniques.

Findings

Curvature is encoded in the small-time distortion of directional entropy.

The method recovers scalar and sectional curvature from heat diffusion.

The approach produces a tensor matching the classical Riemannian curvature operator.

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.