# Polar actions on Hermitian and quaternion-Kähler symmetric spaces

@article{Tebege2006PolarAO, title={Polar actions on Hermitian and quaternion-K{\"a}hler symmetric spaces}, author={Samuel Tebege}, journal={Geometriae Dedicata}, year={2006}, volume={129}, pages={155-171} }

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these…

## 7 Citations

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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.

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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.

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We study polar actions with horizontal sections on the total space of certain principal bundles G/K → G/H with base a symmetric space of compact type. We classify such actions up to orbit equivalence…

## References

SHOWING 1-10 OF 72 REFERENCES

### A classification of hyperpolar and cohomogeneity one actions

- Mathematics
- 2001

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…

### Polar actions on symmetric spaces

- Mathematics
- 2005

We study isometric Lie group actions on symmetric spaces admitting a section, i.e. a submanifold which meets all orbits orthogonally at every intersection point. We classify such actions on the…

### Polar Actions on Compact Symmetric Spaces Which Admit a Totally Geodesic Principal Orbit

- Mathematics
- 2004

We prove that an effective polar action of a compact Lie group on an irreducible symmetric space of compact type admitting a nontrivial totally geodesic principal orbit is the standard action of an…

### Differential Geometry and Symmetric Spaces

- Mathematics
- 1962

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric…

### Polar coordinates induced by actions of compact Lie groups

- Mathematics
- 1985

Let G be a connected Lie subgroup of the real orthogonal group 0(n). For the action of G on R', we construct linear subspaces a that intersect all orbits. We determine for which G there exists such…

### Symplectically asystatic actions of compact Lie groups

- Mathematics
- 2006

In this paper we introduce the notion of symplectically asystatic Hamiltonian action on a Kahler manifold. In the algebraic setting we prove that if a complex linear group G acts on a Kahler manifold…

### QUATERNIONIC KAHLER 8-MANIFOLDS WITH POSITIVE SCALAR CURVATURE

- Mathematics
- 2008

Consider the subgroup of SO(4n) consisting of unit quaternions acting on H = R by right multiplication. We denote the normalizer of this subgroup by Sp(n)Sp(l), which is a maximal subgroup of SO(4n)…

### Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

- Mathematics
- 2004

We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.

### Convexity and Commuting Hamiltonians

- Mathematics
- 1982

The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie…

### Coisotropic actions on compact homogeneous Kähler manifolds

- Mathematics
- 2003

Abstract. The main result of the paper is that a compact homogeneous Kähler manifold admitting an isometric and coisotropic action with a fixed point is isometric to a Hermitian symmetric space.