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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
DOI: https://doi.org/10.1090/S0002-9947-01-02803-3
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.


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Additional Information

Andreas Kollross
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
Email: kollross@math.uni-augsburg.de

DOI: https://doi.org/10.1090/S0002-9947-01-02803-3
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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