We show that the first five of the axioms we had formulated on spectral triples suffice (in a slightly stronger form) to characterize the spectral triples associated to smooth compact manifolds. The algebra, which is assumed to be commutative, is shown to be isomorphic to the algebra of all smooth functions on a unique smooth oriented compact manifold, while the operator is shown to be of Dirac type and the metric to be Riemannian.
Cite this article
Alain Connes, On the spectral characterization of manifolds. J. Noncommut. Geom. 7 (2013), no. 1, pp. 1–82