×

The Poisson-Nijenhuis manifolds revisited. (English) Zbl 0852.58042

Author’s abstract: “We give a short exposition of the basic results of the theory of Poisson-Nijenhuis manifolds developed by M. Magri and C. Morosi [‘A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds’, Quaderno S. 19, Univ. Milan (1984)] and by Y. Kosmann-Schwarzbach and F. Magri, Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 1, 35-81 (1990; Zbl 0707.58048)], using Lie algebroids and noticing a certain generalization. Then, we consider affine Poisson structures of cotangent bundles \(T^*M\) and show that these structures are associated with a Lie algebroid structure of \(TM\) and a 2-form of \(M\). We examine the case where the affine Poisson structure is compatible with the canonical symplectic structure of \(T^*M\), and thereby it provides \(T^*M\) with a Poisson-Nijenhuis structure”.
Reviewer: W.Mozgawa (Lublin)

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58H15 Deformations of general structures on manifolds
17B99 Lie algebras and Lie superalgebras

Citations:

Zbl 0707.58048
PDFBibTeX XMLCite