Annals of Global Analysis and Geometry

, Volume 13, Issue 1, pp 91–98 | Cite as

On the curvatures of Einstein spaces

  • Hyoungsick Bahn
  • Sungpyo Hong


For a pseudo-Riemannian manifold (M, g) of dimensionn≥3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofTpM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.

Key words

Scalar curvature Pseudo-Riemannian manifolds Einstein space 

MSC 1991

53B20 53C50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Dajczer, M.;Nomizu, K.: On sectional curvature of indefinite metrics II.Math. Ann. 247 (1980), 279–282.Google Scholar
  2. [2]
    Dajczer, M.;Nomizu, K.: On the boundedness of Ricci curvature of an indefinite metric.Bol. Soc. Bras. Mat. 11 (1980), 25–30.Google Scholar
  3. [3]
    Graves, L.;Nomizu, K.: On sectional curvature of indefinite metrics.Math. Ann. 232 (1978), 267–272.Google Scholar
  4. [4]
    Harris, S.: A triangle comparison theorem for Lorentz manifolds.Indiana Univ. Math. J. 31 (1982), 289–308.Google Scholar
  5. [5]
    Kulkarni, R.: The values of sectional curvature in indefinite metrics,Comm. Math. Helv. 54 (1979), 173–176.Google Scholar
  6. [6]
    O'Neill, B.:Semi-Riemannian geometry. Academic Press, New York 1983.Google Scholar
  7. [7]
    Schouten, J.;Struik, D.: On some properties of general manifolds relating to Einstein's theory of gravitation.Amer. J. Math. 43 (1921), 213–216.Google Scholar
  8. [8]
    Singer, I.;Thorpe, J.: The curvature of 4-dimensional Einstein spaces. In:Global Analysis, Papers in Honor of K. Kodaira. Princeton University Press, Princeton 1969, pp. 355–365.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Hyoungsick Bahn
    • 1
  • Sungpyo Hong
    • 1
  1. 1.Department of MathematicsPohang University of Science and TechnologyPohangKorea

Personalised recommendations