Geometriae Dedicata

, Volume 108, Issue 1, pp 141–152 | Cite as

Integrable Riemannian Submersion with Singularities

  • Marcos M. Alexandrino


A map of a Riemannian manifold into an euclidian space is said to be transnormal if its restrictions to neighbourhoods of regular level sets are integrable Riemannian submersions. Analytic transnormal maps can be used to describe isoparametric submanifolds in spaces of constant curvature and equifocal submanifolds with flat sections in simply connected symmetric spaces. These submanifolds are also regular leaves of singular Riemannian foliations with sections. We prove that regular level sets of an analytic transnormal map on a real analytic complete Riemannian manifold are equifocal submanifolds and leaves of a singular Riemannian foliation with sections.

singular Riemannian foliations isoparametric maps Riemannian submersions equifocal submanifolds 


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  1. 1.
    Alexandrino, M. M.: Folheações Riemannianas Singulares com Seções e Aplicações Transnormais, PhD thesis, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Setembro de 2002Google Scholar
  2. 2.
    Cartan, E.: Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. di Mat. 17 (1938), 177–191.MATHGoogle Scholar
  3. 3.
    Cartan, E.: Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces shériques, Math. Z. 45 (1939), 335–367.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Cartan, É.: Sur quelques familles remarquables d'hypersurfaces, C. R. Congrès Math. Liège. (1939), 30–41.Google Scholar
  5. 5.
    Cartan, E.: Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions, Revista Univ. Tucumán 1 (1940), 5–22.MATHMathSciNetGoogle Scholar
  6. 6.
    Carter, S. and West, A.: Isoparametric systems and transnormality, Proc. London Math. Soc. 51 (1985), 520–542.MathSciNetGoogle Scholar
  7. 7.
    Carter, S. and West, A.: Generalised Cartan polynomials, J. London Math. Soc. 32 (1985), 305–316.MathSciNetGoogle Scholar
  8. 8.
    Harle, C. E.: Isoparametric families of submanifolds, Bol. Soc. Brasil. Mat. 13 (1982), 491–513.MathSciNetGoogle Scholar
  9. 9.
    Heintze, E., Liu X. and Olmos, C.: Isoparametric submanifolds and a Chevalley-Type restriction theorem. Preprint (2000),, math.dg.0004028.Google Scholar
  10. 10.
    Molino, P.: Riemannian Foliations, Progr. in Math. 73, Birkäuser, Boston, 1988.Google Scholar
  11. 11.
    Palais, R. S. and Terng, C. L.: Critical Point Theory and Submanifold Geometry, Lecture Notes in Math. 1353, Springer-Verlag, Berlin, 1988.Google Scholar
  12. 12.
    Terng, C.-L.: Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), 79–107.MATHMathSciNetGoogle Scholar
  13. 13.
    Terng, C.-L. and Thorbergsson, G.: Submanifold geometry in symmetric spaces. J. Differential Geom. 42 (1995), 665–718.MathSciNetGoogle Scholar
  14. 14.
    Thorbergsson, G.: A Survey on Isoparametric Hypersurfaces and their Generalizations, Handbook of Differential Geometry 1, Elsevier Science, Amsterdam, 2000.Google Scholar
  15. 15.
    Wang, Q.-M.: Isoparametric functions on Riemannian manifolds. I. Math. Ann. 277 (1987), 639–646.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Marcos M. Alexandrino
    • 1
  1. 1.Dept de MatemáticaPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

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