On Noncommutative and semi-Riemannian Geometry

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.

27 pages, LaTeX, fixed typos, changed signature of the metric to properly include the Riemannian case, one reference added

References in corpus (2)

• Spectral Quadruples
• Poincare Duality and Spin^c Structures for Noncommutative Manifolds

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