Abstract
In the sixties V.I. Arnold made a conjecture that (under certain assumptions) the number of intersection points of a Lagrangian submanifold with its image under an arbitrary Hamiltonian isotopy is not less than the sum of its Betti numbers. This conjecture became one of the main forces stimulating the development of Symplectic Topology. Though the complete solution is not yet found, the confirmative answer was obtained in various important cases. Omitting here the historical survey (see e.g. [McD2]) we mention only that one of the most remarkable contributions was made by Andreas Floer [F].
A preliminary of this paper appeared as a preprint (IHES/M/91/61, September 1991) entitled «Transversal Hamiltonian fields and differential relations».
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References
V.I. Arnold, Mathematical methods of classical mechanics, Grundlehren 250, Springer, Berlin-Heidelberg-New-York, 1978.
V.I. Arnold, A.B. Givental, «Symplectic geometry», in Dynamical systems 4, Encyclopedia of Math. Sciences, Springer, 1990, 1–136.
R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, tensor analysis and applications, Springer, Berlin-Heidelberg-New-York, 1988.
A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geometry 28 (1988), 513–547.
M. Gromov, Partial differential relations. Springer Ergebnisse 9, 1986.
H. Hofer, On the topological properties of symplectic maps, Proceedings of the Royal Society of Edinburgh, 115A (1990), 25–38.
D. McDuff, Application of convex integration to symplectic and contact geometry, Ann. Inst. Fourier 37 (1987), 107–133.
D. McDuff, Elliptic methods in Symplectic Geometry, Bulletin A.M.S 23 (1990), 311–358.
L. Polterovich, Symplectic displacement energy for Lagrangian subman-ifolds, Ergod. Th. & Dynam. Syst. 13 (1993), 357–367.
J.-C. Sikorav, Quelques propriétés des plongements Lagrangiennes, Preprint, Orsay (1990).
J.-C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, ETS Editrice, Pisa, 1990.
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© 1995 Birkhäuser Verlag
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Polterovich, L. (1995). An obstacle to non-Lagrangian intersections. In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E. (eds) The Floer Memorial Volume. Progress in Mathematics, vol 133. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9217-9_24
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DOI: https://doi.org/10.1007/978-3-0348-9217-9_24
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