Spectral Functionals, Nonholonomic Dirac Operators, and Noncommutative Ricci Flows

TL;DR

This paper develops a noncommutative extension of Ricci flow theory using spectral functionals derived from nonholonomic Dirac operators, linking noncommutative geometry with geometric flow analysis.

Contribution

It introduces a spectral functional framework for noncommutative Ricci flows, generalizing Perelman's functionals within nonholonomic spectral triples.

Findings

Spectral averaged energy and entropy functionals are formulated.

Conditions for describing nonholonomic Riemannian configurations are derived.

A connection between spectral functionals and classical Ricci flow is established.

Abstract

We formulate a noncommutative generalization of the Ricci flow theory in the framework of spectral action approach to noncommutative geometry. Grisha Perelman's functionals are generated as commutative versions of certain spectral functionals defined by nonholonomic Dirac operators and corresponding spectral triples. We derive the formulas for spectral averaged energy and entropy functionals and state the conditions when such values describe (non)holonomic Riemannian configurations.