Diffusion determines the compact manifold

TL;DR

This paper proves that the heat diffusion process uniquely characterizes the geometry of compact Riemannian manifolds, establishing a correspondence between heat semigroup isomorphisms and manifold isomorphisms.

Contribution

It provides a concise proof that heat semigroup isomorphisms imply manifold isomorphisms, confirming diffusion's role in determining manifold structure.

Findings

Heat semigroup isomorphisms imply manifold isomorphisms

Diffusion uniquely determines the manifold's geometry

Short proof simplifies previous results

Abstract

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.