Siberian Advances in Mathematics

, Volume 24, Issue 3, pp 187–192 | Cite as

Killing vector fields and the curvature tensor of a Riemannian manifold

  • Yu. G. Nikonorov


We find a convenient expression for the value of the covariant curvature 4-tensor of an arbitrary Riemannian manifold on a quadruple of its Killing vector fields. With its use, we in particular obtain a simple deduction of the well-known formula to calculate the sectional curvature of a homogeneous Riemannian space.


Riemannian manifold Killing vector field homogeneous Riemannian space curvature tensor sectional curvature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. I. Arnol’d, “Sur la géométrie différentielle des groupes de Lie de dimension infinite et ses applications à l’hydrodynamique des fluides parfaits,” Ann. Inst. Fourier (Grenoble) 16, 319 (1966).CrossRefzbMATHGoogle Scholar
  2. 2.
    V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989; Springer-Verlag, New York-Heidelberg-Berlin, 1989).CrossRefzbMATHGoogle Scholar
  3. 3.
    V. N. Berestovskiĭ and Yu. G. Nikonorov, “Regular and quasiregular isometric flows on Riemannian manifolds,” Mat. Tr. 10, 3 (2007) [Siberian Adv. Math. 18, 153 (2008)].zbMATHGoogle Scholar
  4. 4.
    V. N. Berestovskiĭ and Yu. G. Nikonorov, “Killing vector fields of constant length on Riemannian manifolds,” Sibirsk. Mat. Zh. 49, 497 (2008) [Siberian Math. J. 49, 395 (2008)].Google Scholar
  5. 5.
    V. N. Berestovskiĭ and Yu. G. Nikonorov, “Clifford-Wolf homogeneous Riemannian manifolds,” J. Differential Geom. 82, 467 (2009).zbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Berger, A Panoramic View of Riemannian Geometry (Springer-Verlag, Berlin, 2003).CrossRefzbMATHGoogle Scholar
  7. 7.
    A. L. Besse, Einstein Manifolds (Springer-Verlag, Berlin, 1987; Mir, Moscow, 1990).CrossRefzbMATHGoogle Scholar
  8. 8.
    J. Cheeger J. and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, Vol. 9. (North-Holland Publishing Co., Amsterdam-Oxford and American Elsevier Publishing Co., Inc.; New York, 1975).zbMATHGoogle Scholar
  9. 9.
    G. R. Jensen, “The scalar curvature of left-invariant Riemannian metrics,” Indiana Univ. Math. J. 20, 1125 (1970/1971).CrossRefMathSciNetGoogle Scholar
  10. 10.
    W. Killing, “Über die Grundlagen der Geometrie,” J. Reine Angew. Math. 109, 121 (1892).zbMATHGoogle Scholar
  11. 11.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, II. (Interscience Publishers, New York-London-Sydney, 1963, 1969; Nauka, Moscow, 1981).Google Scholar
  12. 12.
    O. Kowalski O. and J. Szenthe, “On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,” Geom. Dedicata 81, 209 (2000); correction: Geom. Dedicata. 84, 331 (2001).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    J. Milnor, “Curvatures of left-invariant metrics on Lie groups,” Adv. Math. 21, 293 (1976).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    B. O’Neill B., “The fundamental equations of a submersion,” Michigan Math. J. 13, 459 (1966).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    K. Nomizu, “Invariant affine connections on homogeneous spaces,” Amer. J. Math. 76, 33 (1954).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Allerton Press, Inc. 2014

Authors and Affiliations

  1. 1.Southern Mathematical Institute of the Vladikavkaz Scientific Center of RAS and the Government of the Republic of North Ossetia-AlaniaVladikavkazRussia

Personalised recommendations