Reduction of Jacobi-Nijenhuis manifolds
The notion of Jacobi-Nijenhuis structure was first introduced by \textit{J. C. Marrero, J. Monterde} and \textit{E. Padron} in [C. R. Acad. Sci., Paris, Sér. I, Math. 329, 797-802 (1999; Zbl 0949.37056)] and refined by the authors in [J. Geom. Phys., to appear]. The authors establish a reduction theorem for Jacobi-Nijenhuis manifolds, and study the relation to the reduction of homogeneous Poisson-Nijenhuis manifolds. Since this notion generalizes the concept of Poisson-Nijenhuis structures and is closely related to Jacobi structures, they start with recalling the reduction procedure for Jacobi manifolds from [\textit{K. Mikami}, Proc. Am. Math. Soc. 105, 686-691 (1989; Zbl 0668.53017) and the authors, C. R. Acad. Sci., Paris, Sér. I 308, 101-103 (1989; Zbl 0665.58012)]. Then they give a reduction procedure for homogeneous Poisson-Nijenhuis manifolds, by imitating the Poisson-Nijenhuis reduction theorem of \textit{I. Vaisman} in [J. Geom. Phys. 19, 90-98 (1996; Zbl 0853.58057)]. The remaining part of the paper is devoted to Jacobi-Nijenhuis manifolds. After recalling the definitions needed, the authors prove a reduction theorem for Jacobi-Nijenhuis structures. This is the main result. It consists essentially in defining, on a submanifold \(S\) of a Jacobi-Nijenhuis manifold \(M\), a foliation so that the manifold \(\widehat{S}\) of the leaves is also a Jacobi-Nijenhuis manifold; the reduced manifold \( \widehat{S} \) being said to be obtained by reduction via \(S\). The reduction of conformally equivalent Jacobi-Nijenhuis structures and the relation between Jacobi-Nijenhuis and homogeneous Poisson-Nijenhuis reduction are then also studied. Finally, two examples of Jacobi-Nijenhuis reduction under a Lie group action \(\Phi\) on a Jacobi manifold \(M\) are produced. In the first example, the reduced manifold is the space \(\widehat{M}\) of the orbits of \(\Phi\), obtained by reduction via \(M\). In the second, the action \(\Phi\) is supposed to admit a momentum map \(J\) and the reduced manifold is a quotient space of a level set of this mapping~\(J\).
scientific article; zbMATH DE number 1758083
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