Metrizability of the Lie algebroid generalized tangent bundle and (generalized) Lagrange $(\rho,\eta)$-spaces

TL;DR

This paper introduces a new class of metrizable vector bundles within the framework of generalized Lie algebroids, and develops generalized Lagrange and Finsler spaces, extending classical results to this broader setting.

Contribution

It presents a novel class of metrizable vector bundles and generalized Lagrange spaces based on Lie algebroids, extending classical geometric structures.

Findings

New example of metrizable vector bundle from Lie algebroid tangent bundle

Introduction of generalized Lagrange ( ho, abla ext{?})-spaces and Finsler spaces

Recovery of classical results when morphisms are identities

Abstract

A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new class of Lagrange spaces, called by use, generalized Lagrange (\rho?;\eta?)-space, Lagrange (\rho?;\eta?)-space and Finsler (\rho?;\eta?)-space are presented. In the particular case of Lie algebroids, new and important results are presented. In particular, if all morphisms are identities morphisms, then the classical results are obtained.