Abstract
We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic.
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Communicated by Y. Kawahigashi
Francqui fellow of the Belgian American Educational Foundation.
Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.
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Stiénon, M., Xu, P. Poisson Quasi-Nijenhuis Manifolds. Commun. Math. Phys. 270, 709–725 (2007). https://doi.org/10.1007/s00220-006-0168-0
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DOI: https://doi.org/10.1007/s00220-006-0168-0