Abstract
We show the nonpositivity of the Einstein–Hilbert action for conformal flat Riemannian metrics on noncommutative 4-torus. In addition, we prove that this 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 4-torus. We also give a new proof of their result. Furthermore, computations of the gradient flow and the scalar curvature of noncommutative 4-torus are given.
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