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Commutative Geometries are Spin Manifolds

A. Rennie
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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.

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A. Rennie. 1999-10-28. Commutative Geometries are Spin Manifolds. https://arxiv.org/abs/math-ph/9903021

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