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Periodic Orbits for Generalized Gradient Flows

  • Sol Schwartzman (a1)

Abstract

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.

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Copyright

References

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1. Brock Fuller, F., The Existence of Periodic Points, Ann. of Math. (2) 57(1953), 229230.
2. Schwartzman, S., Asymptotic Cycles, Ann. of Math. (2) 66(1957), 270284.
3. Schwartzman, S., Global Cross Sections of Compact Dynamical Systems, Proc. Nat. Acad. Sci. (5) 48(1962), 786 791.
4. Schwartzman, S., Parallel Vector Fields and Periodic Orbits, Proc. Amer. Math. Soc. (1) 44(1974), 167168.
5. Tischler, D., On Fibering Certain Foliated Manifolds Over Sl, Topology 9(1970), 153154.
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Periodic Orbits for Generalized Gradient Flows

  • Sol Schwartzman (a1)

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