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**Polar actions on symmetric spaces.**
*(English)*
Zbl 1139.53025

Let \(G\) be a compact group acting isometrically on a Riemannian manifold \(M\). The action of \(G\) on \(M\) is called polar if \(M\) admits an immersed, connected submanifold \(\Sigma\) (called a section) that intersects the orbits ortogonally and meets every orbit. A hyperpolar action is a polar one, such that the metric induced on \(\Sigma\) is flat.

J. Dadok [Trans. Am. Math. Soc. 288, 125–137 (1985; Zbl 0565.22010)] obtained the classification of polar actions on spheres and real projective spaces. F. Podesta and G. Thorbergsson [J. Differ. Geom. 53, 131–175 (1999; Zbl 1040.53071)] classified polar actions on the other compact rank-one symmetric spaces.

In the paper under review, the author classifies polar actions on the compact symmetric spaces with simple isometry group and rank greater that one. More precisely, let \(M\) be a compact symmetric space of rank greater than one whose isometry group \(G\) is simple, and let \(H\subset G\) be a closed connected non-trivial subgroup acting polarly on \(M\). Theorem 1 says that in this case the action of \(H\) on \(M\) is hyperpolar and the sections are embedded submanifolds. In Theorem 2, the author obtains the complete classification of connected Lie groups acting polarly without fixed points on the symmetric spaces of higher rank with simple isometry. Corollary 6.2 implies the corresponding classification for actions with fixed points. Theorem 1 and the results of F. Podesta and G. Thorbergsson imply the following: Let \(H\) be a connected compact Lie group acting polarly on a compact symmetric space \(M\) with simple isometry group, then the section \(\Sigma\) of the \(H\)-action on \(M\) is isometric to a flat torus, a sphere or a real projective space (Corollary 1).

J. Dadok [Trans. Am. Math. Soc. 288, 125–137 (1985; Zbl 0565.22010)] obtained the classification of polar actions on spheres and real projective spaces. F. Podesta and G. Thorbergsson [J. Differ. Geom. 53, 131–175 (1999; Zbl 1040.53071)] classified polar actions on the other compact rank-one symmetric spaces.

In the paper under review, the author classifies polar actions on the compact symmetric spaces with simple isometry group and rank greater that one. More precisely, let \(M\) be a compact symmetric space of rank greater than one whose isometry group \(G\) is simple, and let \(H\subset G\) be a closed connected non-trivial subgroup acting polarly on \(M\). Theorem 1 says that in this case the action of \(H\) on \(M\) is hyperpolar and the sections are embedded submanifolds. In Theorem 2, the author obtains the complete classification of connected Lie groups acting polarly without fixed points on the symmetric spaces of higher rank with simple isometry. Corollary 6.2 implies the corresponding classification for actions with fixed points. Theorem 1 and the results of F. Podesta and G. Thorbergsson imply the following: Let \(H\) be a connected compact Lie group acting polarly on a compact symmetric space \(M\) with simple isometry group, then the section \(\Sigma\) of the \(H\)-action on \(M\) is isometric to a flat torus, a sphere or a real projective space (Corollary 1).

Reviewer: Yurii G. Nikonorov (Rubtsovsk)

### MSC:

53C35 | Differential geometry of symmetric spaces |

53C30 | Differential geometry of homogeneous manifolds |