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.

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.

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The notion of ‫ޚ‬ 2 × ‫ޚ‬ 2 -symmetric spaces is a generalization of classical symmetric spaces, where the group ‫ޚ‬ 2 is replaced by ‫ޚ‬ 2 × ‫ޚ‬ 2 . In this article, a classification is given of the ‫ޚ‬ 2 × ‫ޚ‬ 2 -symmetric spaces G/K where G is an exceptional compact Lie group or Spin(8), complementing recent results of Bahturin and Goze. Our results are equivalent to a classification of ‫ޚ‬ 2 × ‫ޚ‬ 2 -gradings on the exceptional simple Lie algebras e 6 , e 7 , e 8 , f 4 , g 2 and so(8).

We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.Comment: 13 page