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.
Cite
@article{arxiv.math-ph/0110001,
title = {On Noncommutative and semi-Riemannian Geometry},
author = {Alexander Strohmaier},
journal= {arXiv preprint arXiv:math-ph/0110001},
year = {2015}
}
Comments
27 pages, LaTeX, fixed typos, changed signature of the metric to properly include the Riemannian case, one reference added