2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162161We 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.Analysis of PDEs58J53; 35P05; 47F05; 35R30; 31C15text