Skip to main content
SpringerLink
Log in

The partial positivity of the curvature in Riemannian symmetric spaces

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, the partial positivity (resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces is determined. From the classifications of abstract root systems and maximal subsystems, the author gives the calculations for symmetric spaces both in classical types and in exceptional types.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Freudenthal, H. and de Vris, H., Linear Lie groups, Pure Appl. Math., 35, Academic Press, New York, 1969.

    MATH  Google Scholar 

  2. Frankel, T., Manifolds with positive curvature, Pacific Jour. Math., 11, 1961, 165–174.

    MATH  MathSciNet  Google Scholar 

  3. Frankel, T., On the fundmental group of minimal submanifolds, Ann. of Math., 83, 1966, 68–73.

    Article  MathSciNet  Google Scholar 

  4. Kenmotsu, K. and Xia, C. Y., Intersections of minimal submanifolds in Riemannian manifold with partially positive curvature, Kodai Math. J., 18, 1995, 242–249.

    Article  MATH  MathSciNet  Google Scholar 

  5. Kenmotsu, K. and Xia, C. Y., Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific Journal Math., 176, 1996, 129–139.

    MathSciNet  Google Scholar 

  6. Knapp, A. W., Lie Groups, Beyond an Introduction, Second Edition, Progress in Mathematics, Vol. 140, Birkhauser, Boston, 2002.

    MATH  Google Scholar 

  7. Wallach, N. R., On maximal subsystems of root systems, Canad. J. Math., 20, 1968, 555–574.

    MATH  MathSciNet  Google Scholar 

  8. Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, A. M. S., Providence, RI, 2001.

    Google Scholar 

  9. Lee, N., On the lower bound of the mean curvature of constant mean curvature hypersurface in non-compact symmetric spaces and manifolds with a pole, Tohoku Math. Jour., 47(2), 1995, 499–508.

    Article  MATH  Google Scholar 

  10. Lee, N., Determination of the partial positivity of the curvature in symmetric spaces, Ann. di Matim. Pura ed Appl., 171, 1996, 107–129.

    Article  MATH  Google Scholar 

  11. Samelson, H., Notes on Lie Algebras, Universitext, Springer-Verlag, New York, 1989.

    Google Scholar 

  12. Shen, Z., On complete manifolds of nonnegative kth-Ricci curvature, Trans. Amer. Math. Soc., 338, 1993, 289–310.

    Article  MATH  MathSciNet  Google Scholar 

  13. Wu, H., Manifolds of partially positive curvature, Indiana Univ. Math. J., 36, 1987, 525–548.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Xusheng Liu

Authors
  1. Xusheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xusheng Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, X. The partial positivity of the curvature in Riemannian symmetric spaces. Chin. Ann. Math. Ser. B 29, 317–332 (2008). https://doi.org/10.1007/s11401-006-0429-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-006-0429-4

Keywords

2000 MR Subject Classification

Search

Navigation