On noncommutative and pseudo-Riemannian geometry |
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Affiliation: | 1. Department of Physics, Babol Branch, Islamic Azad University, Babol, Iran;2. Department of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran |
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Abstract: | We introduce the notion of a pseudo-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of pseudo-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative pseudo-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 pseudo-Riemannian structure in this way. For the noncommutative tori as well as for pseudo-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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