Tight contact structures on solid tori. (English) Zbl 0894.53036

Author’s abstract: “We study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori.
We prove the existence of a \(\mathbb{Z}\)-valued and an \(\mathbb{R}/2\pi\mathbb{Z}\)-valued invariant of a closed solid torus. We call them the self-linking number and the rotation number, respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic.
At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structures on open solid tori.
At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990 (see also Y. Eliashberg [Invent. Math. 98, 623-637 (1989; Zbl 0684.57012)]. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior”.
Reviewer: D.Perrone (Lecce)


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems


Zbl 0684.57012
Full Text: DOI


[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, MIR, Moscow, 1974.
[2] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Nauka, Moscow, 1978.
[3] V. I. Arnold, A. Givental, Symplectic Geometry, Dynamic Systems IV. · Zbl 0780.58016
[4] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. · Zbl 0875.58002
[5] Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87 – 161 (French). · Zbl 0573.58022
[6] Yakov Eliashberg, Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165 – 192 (English, with French summary). · Zbl 0756.53017
[7] Yakov Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991), no. 3, 513 – 520. · Zbl 0733.58011
[8] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623 – 637. · Zbl 0684.57012 · doi:10.1007/BF01393840
[9] Yakov Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 171 – 193. · Zbl 0809.53033
[10] Emmanuel Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), no. 4, 637 – 677 (French). · Zbl 0766.53028 · doi:10.1007/BF02566670
[11] Emmanuel Giroux, Une structure de contact, même tendue, est plus ou moins tordue, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 6, 697 – 705 (French, with English summary). · Zbl 0819.53018
[12] E. Giroux, Classification of tight contact structures on a three-torus, to appear. · Zbl 0969.53044
[13] John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421 – 450. · Zbl 0092.39301 · doi:10.2307/1970192
[14] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515 – 563. · Zbl 0797.58023 · doi:10.1007/BF01232679
[15] Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 643 – 680. · Zbl 0819.58033 · doi:10.1017/S0143385700005277
[16] Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286 – 294. · Zbl 0141.19407
[17] C. L. Siegel, Note on differential equations on the torus, Ann. of Math. 46 (1945), 423-428. · Zbl 0061.19510
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.