On Noncommutative and semi-Riemannian Geometry

TL;DR

This paper extends noncommutative geometry to semi-Riemannian manifolds by introducing semi-Riemannian spectral triples, utilizing Krein spaces, and demonstrating that spectral data can recover geometric features.

Contribution

It introduces semi-Riemannian spectral triples, generalizing spectral triples to include semi-Riemannian manifolds within noncommutative geometry, and adapts the framework to Krein spaces.

Findings

Noncommutative tori can be equipped with semi-Riemannian structures.

Spectral data recovers dimension, signature, and integrals.

Krein spaces replace Hilbert spaces in this setting.

Abstract

We introduce the notion of a semi-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semi-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative semi-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators are Krein-selfadjoint. We show that the noncommutative tori can be endowed with a semi-Riemannian structure in this way. For the noncommutative tori as well as for semi-Riemannian spin manifolds the dimension, the signature of the metric, and the integral of a function can be recovered from the spectral data.