Diffusion determines the manifold

TL;DR

This paper establishes that the structure of a Riemannian manifold can be uniquely determined by the properties of its heat semigroup, linking geometric isomorphism to heat diffusion behavior.

Contribution

It proves that two Riemannian manifolds are isomorphic if an order isomorphism intertwines their Dirichlet heat semigroups, under weak smoothness conditions.

Findings

Manifold isomorphism characterized by heat semigroup intertwining

Weak smoothness condition suffices for the result

Heat diffusion encodes geometric structure

Abstract

We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.