Polar actions on certain principal bundles over symmetric spaces of compact type
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- Proc. Amer. Math. Soc. 139 (2011), 2249-2255 Request permission
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.References
- Jürgen Berndt, Sergio Console, and Carlos Olmos, Submanifolds and holonomy, Chapman & Hall/CRC Research Notes in Mathematics, vol. 434, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1990032, DOI 10.1201/9780203499153
- Jiri Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125–137. MR 773051, DOI 10.1090/S0002-9947-1985-0773051-1
- J. E. D’Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215, iii+72. MR 519928, DOI 10.1090/memo/0215
- Claudio Gorodski, Polar actions on compact symmetric spaces which admit a totally geodesic principal orbit, Geom. Dedicata 103 (2004), 193–204. MR 2034957, DOI 10.1023/B:GEOM.0000013806.62086.72
- 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).
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
- Andreas Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), no. 2, 571–612. MR 1862559, DOI 10.1090/S0002-9947-01-02803-3
- Andreas Kollross, Polar actions on symmetric spaces, J. Differential Geom. 77 (2007), no. 3, 425–482. MR 2362321
- C. Olmos and S. Reggiani, The skew-torsion holonomy theorem and naturally reductive spaces, preprint, 2008.
- Richard S. Palais and Chuu-Lian Terng, Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. MR 972503, DOI 10.1007/BFb0087442
- Fabio Podestà and Gudlaugur Thorbergsson, Polar actions on rank-one symmetric spaces, J. Differential Geom. 53 (1999), no. 1, 131–175. MR 1776093
- Fabio Podestà and Gudlaugur Thorbergsson, Polar and coisotropic actions on Kähler manifolds, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1759–1781. MR 1881015, DOI 10.1090/S0002-9947-02-02902-1
- Silvio Reggiani, On the affine group of a normal homogeneous manifold, Ann. Global Anal. Geom. 37 (2010), no. 4, 351–359. MR 2601495, DOI 10.1007/s10455-009-9190-8
- Samuel Tebege, Polar actions on Hermitian and quaternion-Kähler symmetric spaces, Geom. Dedicata 129 (2007), 155–171. MR 2353989, DOI 10.1007/s10711-007-9202-4
- Wolfgang Ziller, The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helv. 52 (1977), no. 4, 573–590. MR 474145, DOI 10.1007/BF02567391
Additional Information
- Marco Mucha
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010 São Paulo, SP 05508-090, Brazil
- Email: mmuchao@ime.usp.br
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2249-2255
- MSC (2010): Primary 57S15; Secondary 53C35
- DOI: https://doi.org/10.1090/S0002-9939-2011-10868-2
- MathSciNet review: 2775402