A Riemannian homogeneous manifold is said to be a commutative space if the algebra of all isometry-invariant differential operators is a commutative algebra. The Gelʹfand theorem states that this holds for all symmetric spaces but the class of commutative spaces is much broader. In this note the author proves that a complete connected Riemannian manifold such that all isometry-invariant differential operators commute is necessarily a Riemannian homogeneous space. A local analogue of this result has been proved by F. Prüfer, F. Tricerri and the reviewer [Trans. Amer. Math. Soc. {\bf 348} (1996), no.~11, 4643--4652; MR1363946 (97a:53074)] by a completely different method.