Commutative Geometries are Spin Manifolds

TL;DR

This paper provides a detailed proof that Connes' axioms for noncommutative geometry, when applied to the commutative case, characterize spin or spin^c manifolds, extending to pseudo-Riemannian cases.

Contribution

It offers a detailed, elementary proof confirming Connes' claim that commutative noncommutative geometry axioms characterize spin manifolds, including pseudo-Riemannian cases.

Findings

Connes' axioms recover spin or spin^c geometry in the commutative case.

The proof extends to pseudo-Riemannian spin manifolds.

The approach is explicit and elementary.

Abstract

In [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.