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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 \(S^ 3\).

53D35 Global theory of symplectic and contact manifolds
57M50 General geometric structures on low-dimensional manifolds
53D10 Contact manifolds (general theory)
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
Full Text: DOI Numdam EuDML
[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
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