The geometry of symplectic pairs

Bande, G.; Kotschick, D.

doi:10.1090/S0002-9947-05-03808-0

The geometry of symplectic pairs
HTML articles powered by AMS MathViewer

by G. Bande and D. Kotschick PDF

Trans. Amer. Math. Soc. 358 (2006), 1643-1655 Request permission

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

G. Bande, Formes de contact généralisé, couples de contact et couples contacto-symplectiques, Thèse de Doctorat, Université de Haute Alsace, Mulhouse 2000.
Gianluca Bande, Couples contacto-symplectiques, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1699–1711 (French, with English summary). MR 1946411, DOI 10.1090/S0002-9947-02-03209-9
G. Bande, P. Ghiggini, and D. Kotschick, Stability theorems for symplectic and contact pairs, Int. Math. Res. Not. 68 (2004), 3673–3688. MR 2130050, DOI 10.1155/S1073792804141974
G. Bande, A. Hadjar, Contact pairs, Tohoku Math. J. (to appear).
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. MR 112160, DOI 10.2307/1970165
Grant Cairns and Étienne Ghys, Totally geodesic foliations on $4$-manifolds, J. Differential Geom. 23 (1986), no. 3, 241–254. MR 852156
Hansjörg Geiges, Symplectic structures on $T^2$-bundles over $T^2$, Duke Math. J. 67 (1992), no. 3, 539–555. MR 1181312, DOI 10.1215/S0012-7094-92-06721-4
Hansjörg Geiges, Symplectic couples on $4$-manifolds, Duke Math. J. 85 (1996), no. 3, 701–711. MR 1422363, DOI 10.1215/S0012-7094-96-08527-0
Claude Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991 (French). Études géométriques. [Geometric studies]; With a preface by G. Reeb. MR 1120547
Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
M. Hamilton, Bi-Lagrangian structures on closed manifolds, Diplomarbeit München 2004.
Jonathan A. Hillman, Flat $4$-manifold groups, New Zealand J. Math. 24 (1995), no. 1, 29–40. MR 1348051
J. A. Hillman, Four-manifolds, geometries and knots, Geometry & Topology Monographs, vol. 5, Geometry & Topology Publications, Coventry, 2002. MR 1943724
D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29 (1997), no. 2, 145–149. MR 1425990, DOI 10.1112/S0024609396002287
D. Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001), no. 3, 521–531. MR 1828300, DOI 10.1215/S0012-7094-01-10734-5
D. Kotschick and S. Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005), no. 1, 131–149. MR 2104005, DOI 10.1016/j.top.2004.05.002
Vladimir S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), no. 3, 579–609. MR 1961339, DOI 10.1007/s00222-002-0263-6
V. S. Matveev, GF2003 Problem, Kyoto 2003.
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
J. S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46 (1971), 467–477. MR 300307, DOI 10.1007/BF02566859
Koichi Sakamoto and Shinji Fukuhara, Classification of $T^{2}$-bundles over $T^{2}$, Tokyo J. Math. 6 (1983), no. 2, 311–327. MR 732086, DOI 10.3836/tjm/1270213873
Ira H. Shavel, A class of algebraic surfaces of general type constructed from quaternion algebras, Pacific J. Math. 76 (1978), no. 1, 221–245. MR 572981
N. I. Shepherd-Barron, Infinite generation for rings of symmetric tensors, Math. Res. Lett. 2 (1995), no. 2, 125–128. MR 1324696, DOI 10.4310/MRL.1995.v2.n2.a2
Masaaki Ue, Geometric $4$-manifolds in the sense of Thurston and Seifert $4$-manifolds. I, J. Math. Soc. Japan 42 (1990), no. 3, 511–540. MR 1056834, DOI 10.2969/jmsj/04230511
M. Wagner, Über die Klassifikation flacher Riemannscher Mannigfaltigkeiten, Diplomarbeit Basel 1997.
C. T. C. Wall, Geometries and geometric structures in real dimension $4$ and complex dimension $2$, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 268–292. MR 827276, DOI 10.1007/BFb0075230
C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), no. 2, 119–153. MR 837617, DOI 10.1016/0040-9383(86)90035-2