Annals of Global Analysis and Geometry

, Volume 15, Issue 1, pp 51–69 | Cite as

Homogeneity and Canonical Connections of Isoparametric Manifolds

  • Katrin Leschke


On every isoparametric submanifold M a connection with parallel second fundamental form is constructed geometrically such that M is an orbit of an s-representation if and only if the connection is a canonical one. If the rank of M is greater than one this connection is in case of homogeneity the canonical connection of the reductive decomposition given by the orbit of s-representation.

canonical connection homogeneity isoparametric 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abresch, U.: Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann. 264 (1983), 283–302.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Araki, S.: On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. 13(1) (1962), 1–34.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Black, M.: Harmonic Maps into Homogeneous Spaces, Pitman Res. Notes Math. Ser., Vol. 255, Longman Sci. Tech., Harlow, 1991.Google Scholar
  4. 4.
    Cartan, E.: Familles des surfaces isoparamétriques dans les espaces à courbure constante, Ann. di Math. 17 (1938), 177–191.zbMATHCrossRefGoogle Scholar
  5. 5.
    Cartan, E.: Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939), 335–367.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cartan, E.: Sur quelques familles remarquables d'hypersurfaces, C.R. Congrès Math. Liège (1939), 30–41.Google Scholar
  7. 7.
    Cartan, E.: Sur quelques familles remarquables d'hypersurfaces isoparamétriques des espaces sphériques à 5 et 9 dimensions, Rev. Univ. Tucuman, serie A 1 (1940), 5–22.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Dadok, J.: Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), 125–137.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dorfmeister, J. and Neher, E.: Isoparametric hypersurfaces, case g = 6, m = 1, Comm. Algebra 13 (1985), 2299–2368.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Ferus, D., Karcher, H. and Münzner, H. F.: Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), 479–502.zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.Google Scholar
  12. 12.
    Heintze, E., Olmos, C. and Thorbergsson, G.: Submanifolds with constant principal curvatures and normal holonomy groups, Internat. J. Math. 2 (1991), 167–175.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vols. I, II, Interscience, New York, 1963.Google Scholar
  14. 14.
    Levi-Civita, T.: Famiglie di superficie isoparametriche nell'ordinario spacio euclideo, Rend. Acad. Lincei 26(II) (1937), 355–362.zbMATHGoogle Scholar
  15. 15.
    Loos, O.: Symmetric Spaces II: Compact Spaces and Classification, W.A. Benjamin, New York, 1969.Google Scholar
  16. 16.
    Münzner H. F.: Isoparametrische Hyperflächen in Sphären I, II, Math. Ann. 251 (1980), 57–71; 265 (1981), 215–232.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Olmos, C.: Isoparametric submanifolds and their homogeneous structures, J. Diff. Geometry 38 (1993), 225–234.zbMATHMathSciNetGoogle Scholar
  18. 18.
    Olmos, C. and Sanchez, C. U.: A geometric characterization of orbits of s-representations, J. reine angew. Math. 420 (1991), 195–202.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Ozeki, H. and Takeuchi, M.: On some types of isoparametric hypersurfaces in spheres I, II, Tohoku Math. J. 27 (1975), 515–559; 28 (1976), 7–55.zbMATHGoogle Scholar
  20. 20.
    Palais, R. and Terng, C.-L.: A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987), 771–789.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Palais, R. and Terng, C.-L.: Critical Point Theory and Differential Geometry, Lect. Notes Math., Vol. 1353, Springer-Verlag, Berlin, 1988.Google Scholar
  22. 22.
    Segre, B.: Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni, Rend. Accad. Lincei 27 (1938), 203–207.zbMATHGoogle Scholar
  23. 23.
    Terng, C.-L.: Isoparametric submanifolds and their Coxeter groups, J. Differ. Geom. 21 (1985), 79–107.zbMATHMathSciNetGoogle Scholar
  24. 24.
    Terng, C.-L.: Recent progress in submanifold geometry, Proc. Symp. Pure Math. 54 (1992), 439–484.MathSciNetGoogle Scholar
  25. 25.
    Thorbergsson, G.: Isoparametric submanifolds and their buildings, Ann. Math. 133 (1991), 429–446.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Takagi, R. and Takahashi, T.: On the principal curvatures of homogeneous hypersurfaces in the sphere, Diff. Geom., in Honor of Yano (1972), 468–481.Google Scholar
  27. 27.
    Terng, C.-L. and Thorbergsson, G.: Submanifold geometry in symmetric spaces, J. Diff. Geom. 42 (1995), 665–718.zbMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Katrin Leschke
    • 1
  1. 1.Technische Universitäat BerlinBerlinGermany

Personalised recommendations