Dirac-Nijenhuis manifolds
In the paper under review, the notion of Dirac-Nijenhuis manifold is introduced and its relationship to structures studied by \textit{Y.~Kosmann-Schwarzbach} and \textit{F.~Magri} [Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 1, 35--81 (1990; Zbl 0707.58048)] and \textit{F.~Magri} and \textit{C.~Morosi} [A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S 19/1984, Dept. Math. University of Milano, unpublished (1984)] is explained. In order to describe what a Dirac-Nijenhuis structure on a manifold \(P\) is, consider two commuting Nijenhuis tensors \(N_1,N_2\colon TP\to TP\). Then each \(N_j\) is a \((1,1)\)-tensor field satisfying \([N_jX,N_jY]=N_j[N_jX,Y]+N_j[X,N_jY]-N_j^2[X,Y]\) for all vector fields \(X\) and \(Y\) on \(P\). Since \(N_1N_2=N_2N_1=:N\), it follows that \(N\) is also a Nijenhuis tensor, hence it defines a so-called deformed bracket \([\cdot,\cdot]'\) on \(TP\) by \([X,Y]'=[NX,Y]+[X,NY]-N[X,Y]\), and then, \((TP)':=(TP,[\cdot,\cdot]',N)\) is a Lie algebroid. Now, if \(L\) is a Dirac subbundle in both bialgebroids \((TP,T^*P)\) and \(((TP)',T^*P)\), then the triple \((L,N_1,N_2)\) is said to be a Dirac-Nijenhuis structure on \(P\).
scientific article; zbMATH DE number 2077371
scientific article