Mathematische Zeitschrift

, Volume 255, Issue 2, pp 427–436

Singular Riemannian foliations on nonpositively curved manifolds



We prove the nonexistence of a proper singular Riemannian foliation admitting section in compact manifolds of nonpositive curvature. Then we give a global description of proper singular Riemannian foliations admitting sections on Hadamard manifolds. In addition by using the theory of taut immersions we provide a short proof of this result in the special case of a polar action.


Singular Riemannian foliations Nonpositive curvature 

Mathematics Subject Classification (2000)

53C12 57R30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abels H. (1974) Parallelizability of proper actions, global K-slices and maximal compact subgroups. Math. Ann. 212, 1–19MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alekseevsky, D., Kriegl, A., Losik, M., Michor, P.W.: Reflection Groups on Riemannian Manifolds. Preprint 2003, see math.DG/0306078 on (2003)Google Scholar
  3. 3.
    Alexandrino M. (2004) Singular Riemannian foliations with sections. Illinois J. Math. 48(4): 1163–1182MATHMathSciNetGoogle Scholar
  4. 4.
    Bott R., Samelson H. (1958) Applications of the theory of Morse to symmetric spaces. Am. J. Math. 80, 964–1029MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carrière Y. (1984) Les propriétés topologiques des flots riemanniens retrouvées à l’aide du thérème des variétés presque plates. Math. Z. 186, 393–400MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Conlon L. (1971) Variational completeness and K-transversal domains. J. Diff. Geom. 5, 135–147MATHMathSciNetGoogle Scholar
  7. 7.
    Dadok J. (1985) Polar coordinates induced by actions of compact lie groups. Trans. Am. Math. Soc. 288, 125–137MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gutkin E. Geometry and combinatorics of groups generated by reflections. Enseign. Math (2) 32(1–2), 95–110 (1986)Google Scholar
  9. 9.
    Molino P. (1988) Riemannian foliations Progress in Mathematics, vol 73. Birkhäuser, BostonGoogle Scholar
  10. 10.
    Ewert, H.: Equifocal Submanifolds in Riemannian Symmetric Spaces. Doctoral Dissertation, Universität zu Köln (1998)Google Scholar
  11. 11.
    Kollross A. (2002) A classification of hyperpolar and cohomogeneity one actions. Trans. Am. Math. Soc. 354(2): 571–612MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Palais R.S., Terng C.L. (1988) Critical Point Theory and Submanifold Geometry. Geometry Lecture Notes in Math, vol 1353. Springer, Berlin Heidelberg New YorkGoogle Scholar
  13. 13.
    Podestà F., Thorbergsson G. (1999) Polar actions on rank one symmetric spaces. J. Diff. Geom. 53(1): 131–175MATHGoogle Scholar
  14. 14.
    Terng C.L., Thorbergsson G. (1995) Submanifold geometry in symmetric spaces. J. Diff. Geom. 42(3): 665–718MATHMathSciNetGoogle Scholar
  15. 15.
    Thorbergsson, G.: A Survey on Isoparametric Submanifolds and their Generalizations. Handbook of Differential Geometry, vol. I, pp. 963–995. North Holland, Amsterdam (2000)Google Scholar
  16. 16.
    Thorbergsson G. Transformation groups and submanifold geometry. Rend. Mat. Appl. (7) 25(1), 1–16 (2005)Google Scholar
  17. 17.
    Töben, D.: Parallel focal structure and singular Riemannian foliations. Trans. Am. Math. Soc. (to appear) (2006)Google Scholar
  18. 18.
    Walczak P. (1991) On quasi-Riemannian foliations. Ann. Global Geom. 9(1): 83–95MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany

Personalised recommendations