Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

TL;DR

This paper explores Ricci flows of almost Kaehler structures on Lie algebroids, linking geometric evolution equations with thermodynamical concepts and constructing solutions related to Einstein and Finsler structures.

Contribution

It introduces a framework for Ricci flows on Lie algebroids using nonholonomic deformations and thermodynamical analogies, extending geometric analysis to new structures.

Findings

Derived almost Kähler-Ricci evolution equations.

Constructed Ricci soliton configurations for Lie algebroids.

Provided examples of off-diagonal solutions for Einstein and Lagrange-Finsler algebroids.

Abstract

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost K\"ahler - Ricci geometric evolution equations. The second goal is to study how fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic distributions modelling (generalized) Einstein or Finsler - Cartan spaces. Finally, there are provided some examples of generic off-diagonal solutions for Lie algebroid type Ricci solitons and (effective) Einstein and Lagrange-Finsler algebroids.