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Periodic Orbits of Isometric Flows

Published online by Cambridge University Press:  22 January 2016

V. Ozols
V. Ozols*
Affiliation:
University of Washington, Seattle
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Let M be a compact C∞ Riemannian manifold, X a Killing vector field on M, and φt its 1-parameter group of isometries of M. In this, paper, we obtain some basic properties of the set of periodic points of φt. We show that the set of least periods is always finite, and the set P(X, t) of points of M having least period t for the vector field X is a totally geodesic submanifold, with possibly non-empty boundary. Moreover, we show there are at least m geometrically distinct closed geodesic orbits of φt, where m is the number of least periods which are not integral multiples of any other least period.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 48 , December 1972 , pp. 169 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

References

[1] Kobayashi, S., Fixed points of isometries, Nagoya Math. J. 13, pp. 63–68.Google Scholar
[2] Ozols, V., Critical points of the displacement function of an isometry, J. of Differential Geometry 3 No. 4 (1969), pp. 411–432.Google Scholar
[3] Ozols, V., Critical points of the length of a Killing vector field, (to appear).Google Scholar
[4] Yang, C. T., On a problem of Montgomery, Proc. Amer. Math. Soc., 8 (1957), 255–257.Google Scholar