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Anosov flows and non-Stein symplectic manifolds. (English) Zbl 0834.53031
Summary: We simplify and generalize McDuff’s construction of symplectic 4- manifolds with disconnected boundary of contact type in terms of the linking pairing defined on the dual of 3-dimensional Lie algebras. This leads us to an observation that an Anosov flow gives rise to a bi-contact structure, i.e. a transverse pair of contact structures with different orientations, and the construction turns out to work for 3-manifolds which admit Anosov flows with smooth invariant volume. Finally, new examples of bi-contact structures are given and related dynamical problems around bi-contact structures are raised.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37D99 Dynamical systems with hyperbolic behavior
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] D. BENNEQUIN, Topologie symplectique, convexité holomorphe et structures de contact, d’après Y. Eliashberg, D. McDuff et al., Séminaire Bourbaki, 725 (1989-1990). · Zbl 0755.32009
[2] Y. ELIASHBERG, Topological characterization of Stein manifolds of dimension > 2, International J. Math., 1 (1990), 19-46. · Zbl 0699.58002
[3] Y. ELIASHBERG, Contact 3-manifolds twenty years Since J. Martinet’s work, Ann. Inst. Fourier, Grenoble, 42 (1-2) (1991), 165-192. · Zbl 0756.53017
[4] Y. ELIASHBERG and M. GROMOV, Convex symplectic manifolds, Proc. Symp. Pure Math. A.M.S., 52 (2) (1991), 135-162. · Zbl 0742.53010
[5] P. FOULON, Preprint in preparation.
[6] H. GEIGES, Preprints.
[7] E. GHYS, Flots d’Anosov dont LES feuilletages stables sont différentiables, Ann. Scient. École Norm. Sup., 20 (1987), 251-270. · Zbl 0663.58025
[8] S. GOODMAN, Dehn surgery and Anosov flows in proc. geom. dynamics conf., Springer Lecture Notes in Mathematics, 1007 (1983). · Zbl 0532.58021
[9] R.C. GUNNING and H. ROSSI, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs N.J., 1965. · Zbl 0141.08601
[10] M. HANDEL and W.P. THURSTON, Anosov flows on new 3-manifolds, Invent. Math., 59 (1980), 95-103. · Zbl 0435.58019
[11] M. HIRSCH, C. Pugh and M. SHUB, Invariant manifolds, Springer Lecture Notes in Mathematics, 583 (1977). · Zbl 0355.58009
[12] F. LAUDENBACH, Orbites périodiques et courbes pseudo-holomorphes, application à la conjecture de Weinstein en dimension 3, d’après H. Hofer et al., Séminaire Bourbaki, 786 (1993-1994). · Zbl 0853.57013
[13] D. MCDUFF, Examples of simply-connected non-Kählerian manifolds, J. Diff. Geom., 20 (1984), 267-277. · Zbl 0567.53031
[14] D. MCDUFF, Symplectic manifolds with contact type boundaries, Invent. Math., 103 (1991), 651-671. · Zbl 0719.53015
[15] Th. PETERNELL, Pseudoconvexity, the Levi problem and vanishing theorems, Encyclopeadia of Mathematical Sciences, 74, Several complex variables VII, Chapter VIII, Springer-Verlag, Berlin (1994). · Zbl 0811.32011
[16] W.P. THURSTON, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. · Zbl 0324.53031
[17] A. WEINSTEIN, On the hypotheses of Rabinowtz’s periodic orbit theorems, J. Diff. Eq., 33 (1979), 353-358. · Zbl 0388.58020
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