Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T10:33:01.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic Orbits for Generalized Gradient Flows

Published online by Cambridge University Press:  20 November 2018

Sol Schwartzman
Sol Schwartzman*
Affiliation:
University of Rhode Island, Kingston, Rhode Island, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Mn be an n-dimensional compact oriented connected Riemannean manifold. It is proved that either of the following conditions is sufficient to insure that the flow defined by a generalized gradient vector field in Mn has either a stationary point or a periodic orbit:

  • a)Mn is the product of a circle with an (n — 1 ) dimensional manifold of non-zero Euler characteristic.

  • b)The (n — 1) dimensional Stiefel-Whitney class of Mn is different from zero and in addition Mn possesses no one-dimensional 2-torsion.

Keywords

58F25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 1 , 01 March 1995 , pp. 117 - 119
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Brock Fuller, F., The Existence of Periodic Points, Ann. of Math. (2) 57(1953), 229–230.Google Scholar
2. Schwartzman, S., Asymptotic Cycles, Ann. of Math. (2) 66(1957), 270–284.Google Scholar
3. Schwartzman, S., Global Cross Sections of Compact Dynamical Systems, Proc. Nat. Acad. Sci. (5) 48(1962), 786– 791.Google Scholar
4. Schwartzman, S., Parallel Vector Fields and Periodic Orbits, Proc. Amer. Math. Soc. (1) 44(1974), 167–168.Google Scholar
5. Tischler, D., On Fibering Certain Foliated Manifolds Over Sl, Topology 9(1970), 153–154.Google Scholar