Legendre transformation and lifting of multi-vectors

This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle T ∗TM and the tangent bundle of the cotangent bundle TT ∗M of any manifold M. Secondly we show how to use this last isomorphism to construct the lifting of any contravariant tensor field on a manifold M to the tangent bundle TM which generalizes the classical lifting of vector fields. We also show that, in the antisymmetric case, this lifting respects the Schouten bracket. This gives a new proof of a recent result of Crainic and Moerdijk. Finally we give an application to the study of the stability of singular points of Poisson manifold and Lie algebroids.