2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58300In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible.Re-tex to get references right. This is a revised version of a previously incorrect version. Changes to the central portion of proof are extensive. 48 ppMathematical PhysicsDifferential GeometryFunctional Analysistext