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Transactions of the American Mathematical Society

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The geometry of symplectic pairs


Authors: G. Bande and D. Kotschick
Journal: Trans. Amer. Math. Soc. 358 (2006), 1643-1655
MSC (2000): Primary 53C15, 57R17, 57R30; Secondary 53C12, 53D35, 58A17
DOI: https://doi.org/10.1090/S0002-9947-05-03808-0
Published electronically: June 21, 2005
MathSciNet review: 2186990
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the geometry of manifolds carrying symplectic pairs consisting of two closed $2$-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.


References [Enhancements On Off] (What's this?)

  • 1. G. Bande, Formes de contact généralisé, couples de contact et couples contacto-symplectiques, Thèse de Doctorat, Université de Haute Alsace, Mulhouse 2000.
  • 2. G. Bande, Couples contacto-symplectiques, Trans. Amer. Math. Soc. 355 (2003), 1699-1711. MR 1946411 (2003j:53118)
  • 3. G. Bande, P. Ghiggini, D. Kotschick, Stability theorems for symplectic and contact pairs, Int. Math. Res. Not. 2004:68 (2004), 3673-3688. MR 2130050
  • 4. G. Bande, A. Hadjar, Contact pairs, Tohoku Math. J. (to appear).
  • 5. W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin 1984. MR 0749574 (86c:32026)
  • 6. W. M. Boothby, H. C. Wang, On contact manifolds, Ann. of Math. 68 (1958), 721-734. MR 0112160 (22:3015)
  • 7. G. Cairns, E. Ghys, Totally geodesic foliations on $4$-manifolds, J. Differential Geometry 23 (1986), 241-254. MR 0852156 (87m:53043)
  • 8. H. Geiges, Symplectic structures on $T^{2}$-bundles over $T^{2}$, Duke Math. J. 67 (1992), 539-555. MR 1181312 (93i:57036)
  • 9. H. Geiges, Symplectic couples on $4$-manifolds, Duke Math. J. 85 (1996), 701-711. MR 1422363 (98a:53047)
  • 10. C. Godbillon, Feuilletages, Birkhäuser Verlag 1991. MR 1120547 (93i:57038)
  • 11. R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995), 527-595. MR 1356781 (96j:57025)
  • 12. M. Hamilton, Bi-Lagrangian structures on closed manifolds, Diplomarbeit München 2004.
  • 13. J. A. Hillman, Flat $4$-manifold groups, New Zealand J. Math. 24 (1995), 29-40. MR 1348051 (96f:57019)
  • 14. J. A. Hillman, Four-manifolds, geometries and knots, Geometry $\&$ Topology Monographs, vol. 5, 2002. MR 1943724 (2003m:57047)
  • 15. D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29 (1997), 145-149. MR 1425990 (97k:32047)
  • 16. D. Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001), 521-531. MR 1828300 (2002c:53076)
  • 17. D. Kotschick, S. Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005), 131-149. MR 2104005
  • 18. V. S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), 579-609. MR 1961339 (2004f:53044)
  • 19. V. S. Matveev, GF2003 Problem, Kyoto 2003.
  • 20. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294.MR 0182927 (32:409)
  • 21. J. S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46 (1971), 467-477. MR 0300307 (45:9353)
  • 22. K. Sakamoto, S. Fukuhara, Classification of $T^{2}$-bundles over $T^{2}$, Tokyo J. Math. 6 (1983), 311-327. MR 0732086 (85h:55021)
  • 23. I. H. Shavel, A class of algebraic surfaces of general type constructed from quaternion algebras, Pacific J. Math. 76 (1978), 221- 245.MR 0572981 (58:28002)
  • 24. N. I. Shepherd-Barron, Infinite generation for rings of symmetric tensors, Math. Res. Letters 2 (1995), 125-128. MR 1324696 (96i:14018)
  • 25. M. Ue, Geometric $4$-manifolds in the sense of Thurston and Seifert $4$-manifolds I, J. Math. Soc. Japan 42 (1990), 511-540. MR 1056834 (91k:57023)
  • 26. M. Wagner, Über die Klassifikation flacher Riemannscher Mannigfaltigkeiten, Diplomarbeit Basel 1997.
  • 27. C. T. C. Wall, Geometries and geometric structures in real dimension $4$ and in complex dimension $2$, in Geometry and Topology, ed. J. Alexander and J. Harer, Springer Lecture Notes in Math. 1167, Springer Verlag 1985.MR 0827276 (87e:57023)
  • 28. C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), 119-153. MR 0837617 (88d:32038)

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Additional Information

G. Bande
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09129 Cagliari, Italy
Email: gbande@unica.it

D. Kotschick
Affiliation: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany
Email: dieter@member.ams.org

DOI: https://doi.org/10.1090/S0002-9947-05-03808-0
Received by editor(s): May 28, 2004
Published electronically: June 21, 2005
Additional Notes: The authors are members of the European Differential Geometry Endeavour (EDGE), Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme
Article copyright: © Copyright 2005 American Mathematical Society

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