Poisson structures on cotangent bundles (Q1415063)

Let \(\pi\colon T^*M\to M\) be the cotangent bundle of a smooth finite-dimensional manifold \(M\), and \({\mathcal P}(T^*M)\) the algebra of all real smooth functions on \(T^*M\) that are polynomial on the fibers of \(T^*M\). In the paper under review, the author initiates a systematic study of the Poisson structures on \({\mathcal P}(T^*M)\) that are compatible with the natural gradation of \({\mathcal P}(T^*M)\). A description of these structures is achieved in proposition 2.7. Next, one characterizes in proposition~3.9 the semi-graded Poisson structures on \(T^*M\) that are \(\pi\)-related to a given Poisson structure on \(M\). In the last part of the paper, one describes a method to construct Poisson structures on \(T^*M\) starting from a Riemannian metric \(g\) on \(M\), a 2-form on \(M\) and a pseudo-Riemannian metric on \(T^*M\) related to \(g\). The construction naturally involves the Liouville form on \(T^*M\). To conclude, we mention that the paper under review is closely related to a paper on Poisson structures on tangent (rather than cotangent) bundles by \textit{G.~Mitric} and \textit{I.~Vaisman} [Differ. Geom. Appl. 18, No.2, 207-228 (2003; Zbl 1039.53091)].