Poisson Structures on Cotangent Bundles

We make a study of Poisson structures of TM which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poisson structure from M to TM via connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure on TM. Finally, for a 2-form on a Riemannian manifold, we study the conditions for some associated 2-forms on TM to define Poisson structures on cotangent bundles.