Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori

TL;DR

This paper investigates the Einstein-Hilbert action and Gauss-Bonnet theorem in the setting of noncommutative tori, revealing conditions for flatness and computing geometric flows and curvature.

Contribution

It extends previous results by analyzing the Einstein-Hilbert action and Gauss-Bonnet theorem for noncommutative tori with new metric classes and explicit computations.

Findings

Non-positivity of Einstein-Hilbert action for conformal flat metrics

Action vanishes only for constant flat metrics

Gauss-Bonnet theorem established for certain non-diagonal metrics

Abstract

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.