Abstract
Using the covariant derivative for exterior forms of a (dual) vector bundle, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical lifts and between complete lifts are presented. Finally, a duality between Lie algebroids structures is developed
Highlights
The Sasaki lift of a Riemannian metric structure on M is an important example of metric structures on T M which is used in di¤erential geometry with many applications in physics [19]
Lifts of geometrical structures of T M to T T M were introduced and studied by several authors [8, 9, 10, 11, 20]. In many papers such as [5, 12, 14, 17, 18], the authors studied the lifts of geometric objects to the second order tangent bundle, tensor bundle and jet bundle
; ux 2 ua (x) Lar (ux) ; 1 (U ) ; will be called the Legendre transformation of the Lagrangian L: It is remarkable that in the general case, if L is a Lagrange fundamental function on the vector bundle (E; ; M ) and H is its Legendre transformation, H 'L 6= L, but in particular, if L is a Finsler fundamental function on the vector bundle (E; ; M ) and H is its Legendre transformation, H 'L = L: De...nition 5.8
Summary
The Sasaki lift of a Riemannian metric structure on M is an important example of metric structures on T M which is used in di¤erential geometry with many applications in physics [19]. Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality .
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