Legendre transformation and lifting of multi-vectors

Details Legendre transformation and lifting of multi-vectors Legendre transformation and lifting of multi-vectors

2005-02-03

arxiv.org/abs/math/0502068v2

Mathematics

Differential Geometry

This paper has been withdrawn by the author because he has learned that an important part of the results had already been publ

Scientific paper

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.

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