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Poisson Quasi-Nijenhuis Manifolds

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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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Authors and Affiliations

  1. Departement Mathematik, E.T.H. Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Mathieu Stiénon

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Ping Xu

Authors
  1. Mathieu Stiénon
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  2. Ping Xu
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Correspondence to Ping Xu.

Additional information

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

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