Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

TL;DR

This paper introduces a new perspective on the construction of Lie algebroid tangent bundles, defines a covariant derivative for exterior forms, and explores Legendre duality between lifts and Lie algebroid structures.

Contribution

It presents a novel approach to generalized Lie algebroids, introduces a covariant derivative for exterior forms, and develops a theory of Legendre duality for lifts and Lie algebroids.

Findings

Complete lift of sections is characterized using the covariant derivative.

Legendre duality between vertical and complete lifts is established.

A duality between Lie algebroid structures is formulated.

Abstract

Supplementary comments about generalized Lie algebroids are presented and a new point of view over the construction of the Lie algebroid generalized tangent bundle of a (dual) vector bundle is introduced. Using the general theory of exterior differential calculus for generalized Lie algebroids, a covariant derivative for exterior forms of a (dual) vector bundle is introduced. Using this covariant derivative, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical and complete lifts is presented. Finally, a duality between Lie algebroids structures is developed.