  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.
