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Siberian Mathematical Journal

, Volume 49, Issue 3, pp 395–407 | Cite as

Killing vector fields of constant length on Riemannian manifolds

  • V. N. Berestovskii
  • Yu. G. Nikonorov
Article

Abstract

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S 1 on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.

Keywords

Riemannian manifold Killing vector field geodesic Sasaki metric 

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References

  1. 1.
    Killing W., “Über die Grundlagen der Geometrie,” J. Reine Angew. Math., Bd 109, 121–186 (1892).Google Scholar
  2. 2.
    Ricci G. and Levi-Civita T., “Méthodes de calcul différentiel absolu et leurs applications,” Math. Ann., Bd 54, 125–201, 608 (1901).MATHCrossRefGoogle Scholar
  3. 3.
    Eisenhart L. P., Riemannian Geometry, Princeton Univ. Press; Humphrey Milford; Oxford Univ. Press, Princeton; London (1926).MATHGoogle Scholar
  4. 4.
    Bianchi L., Lezioni sulla teoria dei gruppi continui finiti di trasfomazioni, Spoerri, Pisa (1918).Google Scholar
  5. 5.
    Belgun F., Moroianu A., and Semmelmann U., “Symmetries of contact metric manifolds,” Geom. Dedicata, 101, No. 1, 203–216 (2003).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blair D., Contact Manifolds in Riemannian Geometry, Springer-Verlag, Berlin; New York (1976) (Lectures Notes in Math.; 509).MATHGoogle Scholar
  7. 7.
    Boyer C. and Galicki K., “On Sasakian-Einstein geometry,” Internat. J. Math., 11, No. 7, 873–909 (2000).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bott R., “Vector fields and characteristic numbers,” Michigan Math. J., 14, No. 2, 231–244 (1967).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry. Vol. 1 and 2 [Russian translation], Nauka, Moscow (1981).Google Scholar
  10. 10.
    Kobayashi Sh., Transformation Groups in Differential Geometry [Russian translation], Nauka, Moscow (1986).MATHGoogle Scholar
  11. 11.
    Wolf J. A., “Homogeneity and bounded isometries in manifolds of negative curvature,” Illinois J. Math., 8, 14–18 (1964).MATHMathSciNetGoogle Scholar
  12. 12.
    Berger M., “Trois remarques sur les variétés riemanniennes à courbure positive,” C. R. Acad. Sci. Paris Sér. A–B, 263, 76–78 (1966).MATHGoogle Scholar
  13. 13.
    Weinstein A., “A fixed point theorem for positively curved manifolds,” J. Math. Mech., 18, No. 2, 149–153 (1968/1969).MATHMathSciNetGoogle Scholar
  14. 14.
    Wallach N. R., “Compact homogeneous Riemannian manifolds with strictly positive curvature,” Annals of Math., 96, No. 2, 277–295 (1972).CrossRefMathSciNetGoogle Scholar
  15. 15.
    Synge J. L., “On the connectivity of spaces of positive curvature,” Quart. J. Math. Oxford Ser. (1), 7, No. 1, 316–320 (1936).CrossRefGoogle Scholar
  16. 16.
    Yano K. and Bochner S., Curvature and Betti Numbers, Princeton Univ. Press, Princeton, NJ (1953).MATHGoogle Scholar
  17. 17.
    Bochner S., “Vector fields and Ricci curvature,” Bull. Amer. Math. Soc., 52, No. 9, 776–797 (1946).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wadsley A. W., “Geodesic foliations by circles,” J. Differential Geom., 10, No. 4, 541–549 (1975).MATHMathSciNetGoogle Scholar
  19. 19.
    Yang C. T., “On a problem of Montgomery,” Proc. Amer. Math. Soc., 8, No. 2, 255–257 (1957).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Berestovskii V. N. and Nikonorov Yu. G., Killing Vector Fields of Constant Length on Riemannian Manifolds [Preprint arXiv:math.DG/0605371v1 15 May 2006].Google Scholar
  21. 21.
    Montgomery D. and Yang C. T., “On homotopy seven-spheres that admit differentiable pseudo-free circle actions,” Michigan Math. J., 20, No. 3, 193–216 (1973).MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Alekseevsky A. V. and Alekseevsky D. V., “Riemannian G-manifolds with one dimensional orbit space,” Ann. Global Anal. Geom., 11, No. 3, 197–211 (1993).MATHMathSciNetGoogle Scholar
  23. 23.
    Grove K. and Ziller W., “Cohomogeneity one manifolds with positive Ricci curvature,” Invent. Math., 149, No. 3, 619–646 (2002).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Gromoll D., Klingenberg W., and Meyer W., Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).Google Scholar
  25. 25.
    Bangert V., “On the lengths of closed geodesics on almost round spheres,” Math. Z., Bd 191, No. 4, 549–558 (1986).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Tuschmann W., “On the structure of compact simply connected manifolds of positive sectional curvature,” Geom. Dedicata, 67, No. 1, 107–116 (1997).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Besse A. L., Manifolds All of Whose Geodesics Are Closed [Russian translation], Mir, Moscow (1981).Google Scholar
  28. 28.
    Sasaki S., “On the differential geometry of tangent bundles of Riemannian manifolds,” Tohôku Math. J., I: 10, 338–354 (1958); II: 14, 146–155 (1962).MATHCrossRefGoogle Scholar
  29. 29.
    Tanno S., “Killing vectors and geodesic flow vectors on tangent bundles,” J. Reine Angew. Math., Bd 282, 162–171 (1976).MATHMathSciNetGoogle Scholar
  30. 30.
    Bolsinov A. V. and Jovanović B., “Noncommutative integrability, moment map and geodesic flows,” Ann. Glob. Anal. Geom., 23, No. 4, 305–322 (2003).MATHCrossRefGoogle Scholar
  31. 31.
    Bolsinov A. V., “Integrable geodesic flows on Riemannian manifolds,” J. Math. Sci., 123, No. 4, 4185–4197 (2004).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Omsk Branch of the Sobolev Institute of MathematicsOmskRussia
  2. 2.Rubtsovsk Industrial Institute of Altai State Technical UniversityRubtsovskRussia

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