Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



A note on Weinstein's conjecture

Author: Augustin Banyaga
Journal: Proc. Amer. Math. Soc. 109 (1990), 855-858
MSC: Primary 58F22; Secondary 58F05, 58F18
MathSciNet review: 1021206
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the contact foliation of a compact contact manifold $ \left( {M,\alpha } \right)$ has at least one compact leaf in the following two cases: (i) $ \alpha $ is a $ K$-contact form and $ M$ is simply connected, (ii) $ \alpha $ is $ {C^2}$-close to a regular contact form. This solves the Weinstein conjecture in those particular cases.

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

  • [1] D. E. Blair, Contact manifolds in Riemannian geometry, Springer Lecture Notes in Math., vol. 509, Springer-Verlag, New York. MR 0467588 (57:7444)
  • [2] V. L. Ginzburg, New generalizations of Poincaré's geometric theorem, Funct. Anal. Appl. 21 (1987), 100-107. MR 902290 (89f:58057)
  • [3] P. Molino, Riemannian foliations, Progress in Math., Birkhäuser, Boston, 1988, MR 932463 (89b:53054)
  • [4] -, Réduction symplectique et feuilletages Riemanniens: moment structural et théorème de convexité, in Séminaire Gaston Darboux de Géometrie et Topologie Differentielle, 1987-88, Univ. Montpellier, France, pp. 11-25.
  • [5] G. Monna, Feuilletages de $ K$-contact sur les variétés compactes de dimension trois, Pub. Mat. UEB 28 (1984), 81-87. MR 857078 (88d:53030)
  • [6] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure and Appl. Math. 31 (1978), 157-184. MR 0467823 (57:7674)
  • [7] C. Viterbo, A proof of the Weinstein's conjecture inn $ {{\mathbf{R}}^{2n}}$, Ann. Inst. Henri Poincaré 4 (1987), 337-356. MR 917741 (89d:58048)
  • [8] A. Weinstein, On the hypothesis of Rabinowitz' periodic orbit theorem, J. Differential Equations 33 (1978), 353-358. MR 543704 (81a:58030b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F22, 58F05, 58F18

Retrieve articles in all journals with MSC: 58F22, 58F05, 58F18

Additional Information

Keywords: $ K$-contact form, contact foliation, Riemannian foliation, transverse symplectic structure, characteristics
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society