We study perturbations of the flat geometry of the noncommutative two-dimensional torus

$\mathbb {T}^2_\theta$
Tθ2 (with irrational θ). They are described by spectral triples
$(A_\theta , \mathcal {H}, D)$
(Aθ,H,D)
, with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra Aθ of
$\mathbb {T}_\theta$
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.

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