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Kollross, Andreas

A classification of hyperpolar and cohomogeneity one actions. (English) Zbl 1042.53034

Trans. Am. Math. Soc. 354, No. 2, 571-612 (2002).
An isometric action of a compact Lie group on a Riemannian manifold is said to be hyperpolar if there exists a closed connected submanifold that is flat and meets all orbits orthogonally. Examples of hyperpolar actions are the adjoint action of a compact Lie group with a bi-invariant metric, isotropy actions of symmetric spaces and polar representations of compact Lie groups. In [E. Heintze, R. S. Palais, Ch. Terng, and G. Thorbergsson, Hyperpolar actions on symmetric spaces (1995; Zbl 0871.57035)] it was shown that cohomogeneity one actions, i.e. actions whose principal orbits have codimension one, on symmetric spaces of compact type are hyperpolar. The main result in this thorough article are the classification of hyperpolar and cohomogeneity one actions on the irreducible Riemannian symmetric spaces of the compact type (theorems A and B).
In the first section of the article, the author gives all needed definitions and reviews various results in the subject which allow to reduce the classification of irreducible symmetric spaces to actions on simple compact Lie groups. In the second section the author finds those connected subgroups of \(G\times G\), \(G\) a compact simple Lie group, acting hyperpolary on \(G\) and maximal with this property. As a consequence of results in section 2 it follows that all hyperpolar actions that are not Hermann or \(\sigma\)-actions are of cohomogeneity one. In section 3, to complete the classification of cohomogeneity one actions on symmetric spaces, the author determines which Hermann or \(\sigma\)-actions are of cohomogeneity one.
Reviewer: Isabel Dotti Miatello (Córdoba)

Cited in 4 Reviews
Cited in 86 Documents

MSC:

53C35 Differential geometry of symmetric spaces
57S15 Compact Lie groups of differentiable transformations

Keywords:

hyperpolar actions; cohomogeneity one actions; symmetric spaces; compact Lie groups

Citations:

Zbl 0871.57035
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Full Text: DOI

References:

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