Vaisman, I. The Poisson-Nijenhuis manifolds revisited. (English) Zbl 0852.58042 Rend. Semin. Mat., Torino 52, No. 4, 377-394 (1994). 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) Cited in 1 ReviewCited in 60 Documents 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 Keywords:Hamiltonian vector field; symplectic; Lie algebroid; affine Poisson structures; Poisson-Nijenhuis structure Citations:Zbl 0707.58048 PDFBibTeX XMLCite \textit{I. Vaisman}, Rend. Semin. Mat., Torino 52, No. 4, 377--394 (1994; Zbl 0852.58042)