Noncommutative Ricci flow in a matrix geometry

Abstract

We study noncommutative Ricci flow in a finite-dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar curvature in terms of the Ricci flow.

Heat equation in a model matrix geometry

“…The following result is basically proved in Proposition 3.1 in [8] and we refer to [8] for the proof. …”

“…The aim of this paper is to explore the properties of heat equation in a simple case, which has been recently studied by R. Duvenhage in [8]. In [8], the author has introduced the Ricci flow and his main results can be briefly stated as follows.…”

“…The aim of this paper is to explore the properties of heat equation in a simple case, which has been recently studied by R. Duvenhage in [8]. In [8], the author has introduced the Ricci flow and his main results can be briefly stated as follows. Let M n be the C * algebra generated by the two matrices [19] u = e 2π ix/n , v = e 2π i y/n , where x, y are two Hermitian matrices on C n .…”

Analyticity and spectral properties of noncommutative Ricci flow in a matrix geometry

“…In [8], Ricci flow was defined and studied in a simple example of a matrix geometry, namely in a finite dimensional representation of a noncommutative (or quantum) 2-torus. This was motivated by [2], which attempts to define Ricci flow in the usual infinite dimensional representation of the noncommutative 2-torus by using a first variation formula for the eigenvalues of the Laplace-Beltrami operator obtained in the classical case in [7].…”

“…In [8], however, the Ricci flow was defined more directly by a noncommutative version of the Ricci flow equation, with no reference to the spectrum of the Laplace-Beltrami operator or a first variation formula. In this paper the aim is to show that a first variation formula can in fact also be obtained for the Ricci flow as defined in [8].…”