Curved noncommutative torus and Gauss--Bonnet

We study perturbations of the flat geometry of the noncommutative two-dimensional torus T2θ (with irrational θ). They are described by spectral triples (Aθ,H,D) , with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A θ of Tθ . We show, up to the second order in perturbation, that the ζ-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.