  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 to 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.
