  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.

Differential Geometry