On Noncommutative and semi-Riemannian Geometry

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.

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