×

zbMATH — the first resource for mathematics

Contact 3-manifolds twenty years since J. Martinet’s work. (English) Zbl 0756.53017
The paper gives an account of the recent development in 3-dimensional contact geometry. The central result of the paper states that there exists a unique tight contact structure on \(S^ 3\). Together with the earlier classification of overtwisted contact structures on 3-manifolds this result completes the classification of contact structures on \(S^ 3\).

MSC:
53D35 Global theory of symplectic and contact manifolds
57M50 General geometric structures on low-dimensional manifolds
53D10 Contact manifolds, general
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] D. BENNEQUIN, Entrelacements et equations de Pfaff, Astérique, 107-108 (1983), 83-61. · Zbl 0573.58022
[2] J. CERF, Sur LES difféomorphismes de S3 (г = 0), Lect. Notes in Math., 53 (1968). · Zbl 0164.24502
[3] Y. ELIASHBERG, Classification of overtwisted contact structures on 3-manifolds, Invent. Math., 98 (1989), 623-637. · Zbl 0684.57012
[4] Y. ELIASHBERG, The complexification of contact structures on a 3-manifold, Usp. Math. Nauk., 6(40) (1985), 161-162. · Zbl 0601.53029
[5] Y. ELIASHBERG, On symplectic manifolds with some contact properties, J. Diff. Geometry, 33 (1991), 233-238. · Zbl 0735.53021
[6] Y. ELIASHBERG, Filling by holomorphic discs and its applications, London Math. Soc. Lect. Notes Ser., 151 (1991), 45-67. · Zbl 0731.53036
[7] Y. ELIASHBERG, Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math., 1, n°1 (1990), 29-46. · Zbl 0699.58002
[8] Y. ELIASHBERG, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc., 4 (1991), 513-520. · Zbl 0733.58011
[9] Y. ELIASHBERG, Legendrian and transversal knots in tight contact manifolds, preprint, 1991. · Zbl 0809.53033
[10] Y. ELIASHBERG and M. GROMOV, Convex symplectic manifolds, Proc. of Symposia in Pure Math., 52 (1991), part 2, 135-162. · Zbl 0742.53010
[11] E. GIROUX, Convexité en topologie de contact, to appear in Comm. Math. Helvet., 1991. · Zbl 0766.53028
[12] M. GROMOV, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. · Zbl 0592.53025
[13] V. HARLAMOV and Y. ELIASHBERG, On the number of complex points of a real surface in a complex surface, Proc. LITC-82, (1982), 143-148. · Zbl 0609.32016
[14] R. LUTZ, Structures de contact sur LES Fibre’s principaux en cercles de dimension 3, Ann. Inst. Fourier, 27-3 (1977), 1-15. · Zbl 0328.53024
[15] J. MARTINET, Formes de contact sur LES variétés de dimension 3, Lect. Notes in Math, 209 (1971), 142-163. · Zbl 0215.23003
[16] D. MCDUFF, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc., 3, n°1 (1990), 679-712. · Zbl 0723.53019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.