Metric Compatible or Noncompatible Finsler-Ricci Flows

TL;DR

This paper investigates Finsler gravity models and geometric evolution equations involving noncompatible Finsler connections, highlighting the limitations in constructing self-consistent models with arbitrary noncompatible connections.

Contribution

It demonstrates the feasibility of formulating Finsler gravity and Ricci flow models with noncompatible connections using generalized Perelman's functionals.

Findings

Models with metric noncompatible Finsler connections are possible with distortion tensors.

Self-consistent geometric evolution models are generally not feasible with arbitrary noncompatible connections.

The approach extends Ricci flow theory to nonholonomic Finsler geometries.

Abstract

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature etc. In a series of works, we studied (non)commutative metric compatible Finsler and nonholonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The goal of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived nonholonomic Hamilton evolution equations, when metric noncompatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric noncompatible connections.