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

2017/07/15 by Golipour, Javad
#46L87 #Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1707.04777

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.

Related

• Modular Curvature for Noncommutative Two-Tori
• The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure
• Curvature in Noncommutative Geometry
• Asymmetric noncommutative torus has vanishing Einstein tensor
• Scalar Curvature for Noncommutative Four-Tori