  We show the non-positivity of the Einstein-Hilbert action for conformal flat
Riemannian metrics. The action vanishes only when the metric is constant flat.
This recovers an earlier result of Fathizadeh-Khalkhali in the setting of
spectral triples on noncommutative four-torus. Furthermore, computations of the
gradient flow and the scalar curvature of this space based on modular operator
are given. We also show the Gauss-Bonnet theorem for a parametrized class of
non-diagonal metrics on noncommutative two-torus.
