Special symplectic connections and Poisson geometry. (English) Zbl 1070.53050

The authors show how any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure.
The space of symplectic connections is rather huge. Therefore, to study meaningful connections, it is thus necessary to impose further conditions.
In this paper, the authors investigate the so-called special symplectic connections.
They first recall the various conditions of such symplectic connections and their relevance. They also review the standard results and notions related both to symplectic and Poisson geometry, and symplectic groupoids. Then they prove the main result of the paper. Finally, some general consequences are discussed.


53D17 Poisson manifolds; Poisson groupoids and algebroids
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[1] Bourgeois, F. and Cahen, M.: A variational principle for symplectic connections, J. Geom. Phys. 30 (1999), 233-265. · Zbl 0963.53050
[2] Baguis, F. and Cahen, M.: A construction of symplectic connections through reduction, Lett. Math. Phys. 57 (2001), 149-160. · Zbl 1033.53071
[3] Bochner, S.: Curvature and Betti numbers, II, Ann. Math. 50 (1949), 77-93. · Zbl 0039.17603
[4] O ’Brian, N. R. and Rawnsley, J.: Twistor spaces, Ann. Global. Anal. Geom. 3 (1985), 29-58. · Zbl 0564.53030
[5] Bryant, R.: Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Sympos. Pure Math. 53 (1991), 33-88. · Zbl 0758.53017
[6] Bryant, R.: Bochner-Ka ”hler metrics, J. Amer. Math. Soc. 14 (3)((2001), 623-715. · Zbl 1006.53019
[7] Cahen, M., Gutt, S. and Rawnsley, J.: Symmetric symplectic spaces with Ricci-type curvature, In:G. Dito and D. Sternheimer (eds), Confe ’rence Moshe ’Flato 1999, Vol. II, Math. Phys. Stud. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 81-91. · Zbl 0983.53032
[8] Cahen, M., Gutt, S. and Schwachho ”fer, L. J.: Construction of Ricci-type connections by reduction and induction, Preprint, arXiv:math. DG/0310375.
[9] Cahen, M., Gutt, S., Horowitz, J. and Rawnsley, J.: Homogeneous symplectic manifolds with Ricci-type curvature, J. Geom. Phys. 38 (2001), 140-151. · Zbl 0999.53050
[10] Chi, Q.-S., Merkulov, S. A. and Schwachho ”fer, L. J.: On the Existence of In nite Series of Exotic Holonomies, Invent. Math. 126 (1996), 391-411. · Zbl 0866.53013
[11] Cahen, M. and Schwachho ”fer, L. J.: Special symplectic connections, Preprint, arXiv:math. DG/0402221.
[12] Kamishima, Y.: Uniformization of Ka ”hler manifolds with vanishing Bochner tensor, Acta Math. 172 (2)((1994), 299-308. · Zbl 0828.53059
[13] Onishchik, A. L. and Vinberg, E. B.: Lie Groups and Lie Algebras, Vol. 3, Springer-Verlag, Berlin, 1996. · Zbl 0876.22001
[14] Merkulov, S. A. and Schwachho ”fer, L. J.: Classification of irreducible holonomies of torsion free affine connections, Ann. Math. 150 (1999)77-149;Addendum:Classification of irreducible holonomies of torsion-free affine connections, Ann. Math. 150 (1999)1177-1179. · Zbl 0992.53038
[15] Schwachho ”fer, L. J.: On the classification of holonomy representations, Habilitationsschrift, Universita ”t Leipzig (1998).
[16] Schwachho ”fer, L. J.: Homogeneous onnections with special symplectic holonomy, Math. Zeit. 238 (2001), 655-688. · Zbl 1013.53032
[17] Schwachho ”fer, L. J.: Connections with irreducible holonomy representations, Adv. Math. 160 (2001), 1-80.
[18] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkha ”user, Basel, 1994. · Zbl 0810.53019
[19] Vaisman, I.: Variations on the theme of twistor spaces, Balkan J. Geom. Appl. 3 (1998), 135-156. · Zbl 0926.53020
[20] Weinstein, A.: The local structure of Poisson manifolds, J. Differential Geom. 18, (1983)523-557. · Zbl 0524.58011
[21] Xu, P.: Dirac submanifolds and Poisson structures, Preprint, arXiv:math. SG/0110326.
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