2005/02/03 by Jean-Paul Dufour, Dufour, Jean-Paul
Mathematics · Medicine · #53A45 #53D17 #53Z05 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #math.DG #msc:53A45 #msc:53D17 #msc:53Z05
paper · pdf · doi:10.48550/arxiv.math/0502068
This paper has been withdrawn by the author because he has learned that an important part of the results had already been published by J. Grabowski and P. Urbanski (see for example: DG/9710007,9710013,9909174 and ref. herein) A more specific paper on the Legendre transformation is in preparation. Also the last paragraph of our paper will be added to DG/0501168
openalex publication_date 2005/02/03 · arxiv created 2005/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.