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A classification of hyperpolar and cohomogeneity one actions

Author: Andreas Kollross
Journal: Trans. Amer. Math. Soc. 354 (2002), 571-612
MSC (2000): Primary 53C35, 57S15
Published electronically: September 18, 2001
MathSciNet review: 1862559
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Abstract: An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.

References [Enhancements On Off] (What's this?)

  • [BESSE] Besse A. L.: Einstein Manifolds, Springer-Verlag, (1987) MR 88f:53087
  • [BMP] Bremner M.R., Moody R.V., Patera J.: Tables of dominant weight multiplicities for representations of simple Lie algebras, Dekker, New York (1985) MR 86f:17002
  • [CO1] Conlon L.: The topology of certain spaces of paths on a compact symmetric space, Trans. Amer. Math. Soc. 112, 228-248 (1964) MR 29:631
  • [CO2] Conlon L.: Remarks on commuting involutions, Proc. Amer. Math. Soc. 22, 255-257 (1969) MR 39:4873
  • [DADOK] Dadok J.: Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288, 125-137 (1985) MR 86k:22019
  • [D'ATRI] Datri J.E.: Certain isoparametric families of hypersurfaces in symmetric spaces, J. Differential Geom. 14, 21-40 (1979) MR 81g:53041
  • [DYN1] Dynkin E.B.: Semisimple subalgebras of the semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, 6, 111-244 (1952)
  • [DYN2] Dynkin E.B.: The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. Ser. 2, 6, 245-378 (1952)
  • [HL] Heintze E., Liu X.: A splitting theorem for isoparametric submanifolds in Hilbert space, J. Differential Geom. 45, 319-335 (1997) MR 98c:58006
  • [HPTT1] Heintze E., Palais R., Terng C.-L., Thorbergsson G.: Hyperpolar actions on symmetric spaces, Geometry, topology and physics for Raoul Bott, (S.-T. Yau, ed.), International Press, Cambridge, (1995) MR 96i:53052
  • [HPTT2] Heintze E., Palais R., Terng C.-L., Thorbergsson G.: Hyperpolar actions and $k$-flat homogeneous spaces, J. Reine Angew. Math. 454, 163-179 (1994) MR 96b:53062
  • [HE] Helgason S.: Differential geometry, Lie groups and symmetric spaces, Academic Press (1978) MR 80k:53081
  • [HR] Hermann R.: Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11, 544-546 (1960) MR 23:A1748
  • [HSL] Hsiang W.Y., Lawson H.B.: Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5, 1-38 (1971) MR 45:7645
  • [HSHS] Hsiang W.C., Hsiang W.Y.: Differentiable actions of compact connected classical groups II, Ann. of Math. (2) 92, 189-223 (1970) MR 42:420
  • [IWATA] Iwata, K.: Compact transformation groups on rational cohomology Cayley projective planes, Tôhoku Math. J., 33, 429-422 (1981) MR 83h:57047
  • [JÄNICH] Jänich K.: Differenzierbare $G$-Mannigfaltigkeiten, Lecture Notes in Math. 59, Springer-Verlag (1968) MR 37:4835
  • [LM] Lawson H.B. and Michelsohn M.-L.: Spin Geometry, Princeton University Press, Princeton (1989) MR 91g:53001
  • [MANN] Mann L. N.: Gaps in the dimensions of transformation groups, Illinois J. Math. 10, 532-546 (1966) MR 34:282
  • [MU] Murakami S.: Exceptional simple Lie groups and related topics in recent differential geometry, Differential Geometry and Topology, (Jiang B., Peng Ch.-K., Hou Z., eds.), Lecture Notes in Math. 1369, 183-221, Springer (1989) MR 90g:22009
  • [ON1] Oniscik A.L.: Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl. Ser. 2, 50, 5-58 (1966)
  • [ON2] Oniscik A.L.: Topology of transitive transformation groups, Johann Ambrosius Barth, Leipzig (1994)
  • [PT1] Palais R., Terng C.-L.: A general theory of canonical forms, Trans. Amer. Math. Soc. 300, 771-789 (1987) MR 88f:57069
  • [PT2] Palais R., Terng C.-L.: Critical Point Theory and Submanifold Geometry, Lecture Notes in Math. 1353, Springer-Verlag, (1988) MR 90c:53143
  • [PTH] Podestà F., Thorbergsson, G. Polar actions on rank-one symmetric spaces, J. Differential Geom. 53, 131-175 (1999) CMP 2000:16
  • [TAKAGI] Takagi R.: On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10, 495-506 (1973) MR 49:1433
  • [TT] Terng C.-L., Thorbergsson G.: Submanifold geometry in symmetric spaces, J. Differential Geom., 42, (1995) MR 97k:53054
  • [TITS] Tits J.: Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Math. 40, Springer-Verlag (1967) MR 36:1575
  • [WOLF] Wolf J. A.: Spaces of constant curvature, Publish or Perish, Wilmington, Delaware (1977)

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Additional Information

Andreas Kollross
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Keywords: Hyperpolar actions, cohomogeneity one actions, symmetric spaces, compact Lie groups
Received by editor(s): October 10, 2000
Published electronically: September 18, 2001
Additional Notes: Supported by Deutsche Forschungsgemeinschaft
Article copyright: © Copyright 2001 American Mathematical Society

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