Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori
Differential Geometry
2018-07-19 v2 Operator Algebras
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.
Keywords
Cite
@article{arxiv.1707.04777,
title = {Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori},
author = {Javad Golipour},
journal= {arXiv preprint arXiv:1707.04777},
year = {2018}
}