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Polar actions on certain principal bundles over symmetric spaces of compact type

Author: Marco Mucha
Journal: Proc. Amer. Math. Soc. 139 (2011), 2249-2255
MSC (2010): Primary 57S15; Secondary 53C35
Published electronically: February 4, 2011
MathSciNet review: 2775402
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Abstract: We study polar actions with horizontal sections on the total space of certain principal bundles $ G/K\to G/H$ with base a symmetric space of compact type. We classify such actions up to orbit equivalence in many cases. In particular, we exhibit examples of hyperpolar actions with cohomogeneity greater than one on locally irreducible homogeneous spaces with nonnegative curvature which are not homeomorphic to symmetric spaces.

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  • 1. J. Berndt, S. Console, and C. Olmos, Submanifolds and holonomy, Chapman & Hall/CRC, Boca Raton, 2003. MR 1990032 (2004e:53073)
  • 2. J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125-137. MR 773051 (86k:22019)
  • 3. J. E. D'Atri and W. Ziller, Naturally reductive metrics and Eintein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215. MR 0519928 (80i:53023)
  • 4. C. Gorodski, Polar actions on compact symmetric spaces which admit a totally geodesic principal orbit, Geom. Dedicata 103 (2004), no. 1, 193-204. MR 2034957 (2004j:53068)
  • 5. C. Gorodski and G. Thorbergsson, Representations of compact Lie groups and the osculating spaces of their orbits, preprint, University of Cologne, 2000 (also e-print math.DG/0203196).
  • 6. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 2, Interscience Publishers, New York, London, 1969. MR 0238225 (38:6501)
  • 7. A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), no. 2, 571-612. MR 1862559 (2002g:53091)
  • 8. -, Polar actions on symmetric spaces, J. Differential Geom. 77 (2007), no. 3, 425-482. MR 2362321 (2008k:53107)
  • 9. C. Olmos and S. Reggiani, The skew-torsion holonomy theorem and naturally reductive spaces, preprint, 2008.
  • 10. R. S. Palais and C. L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Math., vol. 1353, Springer-Verlag, Berlin, Heidelberg, New York, 1988. MR 972503 (90c:53143)
  • 11. F. Podestà and G. Thorbergsson, Polar actions on rank-one symmetric spaces, J. Differential Geom. 53 (1999), no. 1, 131-175. MR 1776093 (2001j:53064)
  • 12. -, Polar and coisotropic actions on Kähler manifolds, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1759-1781. MR 1881015 (2002j:53096)
  • 13. S. Reggiani, On the affine group of a normal homogeneous manifold, Ann. Global Anal. Geom. 37 (2010), no. 4, 351-359. MR 2601495
  • 14. S. Tebege, Polar actions on Hermitian and quaternion-Kähler symmetric spaces, Geom. Dedicata 129 (2007), no. 1, 155-171. MR 2353989 (2008h:53127)
  • 15. W. Ziller, The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helv. 52 (1977), 573-590. MR 0474145 (57:13795)

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Additional Information

Marco Mucha
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010 São Paulo, SP 05508-090, Brazil

Keywords: Polar action, symmetric space
Received by editor(s): June 2, 2010
Published electronically: February 4, 2011
Additional Notes: This research was supported by FAPESP grant 2007/59288-2.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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