On the spectral characterization of manifolds

TL;DR

This paper demonstrates that a set of five axioms on spectral triples are sufficient to uniquely characterize the spectral triples associated with smooth compact manifolds, linking algebraic and geometric structures.

Contribution

It shows that five specific axioms can fully characterize spectral triples of smooth compact manifolds, establishing a precise algebraic-geometric correspondence.

Findings

Algebra is isomorphic to smooth functions on a unique manifold

Operator is of Dirac type

Metric is Riemannian

Abstract

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.