The noncommutative Lorentzian cylinder as an isospectral deformation

Abstract

We present a new example of a finite-dimensional noncommutative manifold, namely, the noncommutative cylinder. It is obtained by isospectral deformation of the canonical triple associated with the Euclidean cylinder. We discuss Connes’ character formula for the cylinder. In the second part, we discuss noncommutative Lorentzian manifolds. Here, the definition of spectral triples involves Krein spaces and operators on Krein spaces. A central role is played by the admissible fundamental symmetries on the Krein space of square integrable sections of a spin bundle over a Lorentzian manifold. Finally, we discuss isospectral deformation of the Lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder.