Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On characteristics of circle invariant presymplectic forms


Authors: Augustin Banyaga and Philippe Rukimbira
Journal: Proc. Amer. Math. Soc. 123 (1995), 3901-3906
MSC: Primary 53C15; Secondary 58F05
MathSciNet review: 1307491
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a circle-invariant exact 2-form of rank 2n on a compact $ (2n + 1)$-dimensional manifold admits two closed characteristics. This solves a particular case of a generalized Weinstein conjecture.


References [Enhancements On Off] (What's this?)

  • [BAN] Augustin Banyaga, A note on Weinstein’s conjecture, Proc. Amer. Math. Soc. 109 (1990), no. 3, 855–858. MR 1021206, 10.1090/S0002-9939-1990-1021206-0
  • [BAN1] Augustin Banyaga, On characteristics of hypersurfaces in symplectic manifolds, Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 9–17. MR 1216525
  • [BEN] 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). MR 753131
  • [BEN1] -, Quelques remarques simples sur la rigidit é symplectique, Géométrie Symplectique et de Contact: Autour du théorème de Poincaré-Birkhoff, Travaux en Cours (Dazord, Desolneux, and Moulis, eds.), Hermann, Paris, 1984, pp. 1-50.
  • [BLA] David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588
  • [ELI] 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). MR 1162559
  • [HVI] H. Hofer and C. Viterbo, The Weinstein conjecture in cotangent bundles and related results, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 411–445 (1989). MR 1015801
  • [HOV] H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992), no. 5, 583–622. MR 1162367, 10.1002/cpa.3160450504
  • [HOF] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, 10.1007/BF01232679
  • [LUT] Robert Lutz, Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 1, xvii, 283–306 (French, with English summary). MR 526789
  • [RAB] Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 0467823
  • [RUK] Philippe Rukimbira, Some remarks on 𝑅-contact flows, Ann. Global Anal. Geom. 11 (1993), no. 2, 165–171. MR 1225436, 10.1007/BF00773454
  • [VIT] Claude Viterbo, A proof of Weinstein’s conjecture in 𝑅²ⁿ, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 4, 337–356 (English, with French summary). MR 917741
  • [WEI] A. Weinstein, On the hypothesis of Rabinowitz's periodic orbit theorem, J. Differential Equations 33 (1978), 353-358.
  • [ZEH] E. Zehnder, Remarks on periodic solutions on hypersurfaces, Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 209, Reidel, Dordrecht, 1987, pp. 267–279. MR 920629

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C15, 58F05

Retrieve articles in all journals with MSC: 53C15, 58F05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1307491-0
Keywords: Presymplectic form, characteristics, generalized Weinstein conjecture
Article copyright: © Copyright 1995 American Mathematical Society