Lie groupoids and generalized almost paracomplex manifolds

In this paper, we show that there is a close relationship between generalized paracomplex manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized paraholomorphic maps.

Lie groupoids and generalized almost paracomplex manifolds

In this paper, we show that there is a close relationship between generalized paracomplex manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized paraholomorphic maps.

___

  • Bursztyn, H., Crainic, M., Weinstein, A., Zhu, C.: Integration of twisted Dirac brackets, Duke Math. J., 123, 549–607, (2004).
  • Crainic, M.: Generalized complex structures and Lie brackets, Bull. Braz. Math. Soc., New Series 42(4), 559–578, (2011).
  • Ehresmann, C.: Cat`egories topologiques et cat`egories differentiables, Coll. Geom. Diff. Globales, Bruxelles, 137–150, (1959).
  • Gualtieri, M.: Generalized complex geometry, Ph.D. thesis, Univ. Oxford, arXiv:math.DG/0401221, (2003). Hitchin, N.: Generalized Calabi-Yau manifolds, Q. J. Math., 54, 281–308, (2003).
  • Iglesias-Ponte, D., Wade, A.: Contact manifolds and generalized complex structures, J. Geom. Phys., 53, 249–258, (2005).
  • Mackenzie K.: Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, (1987).
  • Pradines, J.: Th´eorie de Lie pour les groupo¨ıdes diff´erentiables, Calcul diff´erentiel dans la cat´egorie des groupo¨ıdes infinit´esimaux, Comptes rendus Acad. Sci. Paris 264 A, 245–248, (1967).
  • Vaisman, I.: Reduction and submanifolds of generalized complex manifolds, Dif. Geom. and its Appl., 25, 147–166, (2007).
  • Vaisman, I.: Lectures on the geometry of Poisson manifolds, Progress in Math., vol. 118, Birkh¨auser Verlag, Boston, (1994).
  • Wade, A.: Dirac structures and paracomplex manifolds, C. R. Acad. Sci. Paris, Ser. I, 338, 889–894, (2004).