Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 57-74.

Special Cohomogeneity One Isometric Actions on Irreducible Symmetric Spaces of Types I and II ^{∗} László Verhóczki Department of Geometry, Eötvös University Pázmány P. s. 1/c, 1117 Budapest, Hungary Abstract. In the present paper we study isometric actions on compact symmet- ic spaces for which the principal orbits are tubular hypersurfaces around totally eodesic singular orbits. We show that in these cases the symmetric space can be hought of as a compact tube the radius of which is determined by the curvature ensor. Since the constant principal curvatures of the tubular orbits can explic- tly be expressed, we obtain a simple method to determine volumes of symmetric paces by using volumes of lower dimensional ones. Finally, we discuss the classical rreducible symmetric spaces of types I and II, each of which admits such special yperpolar actions. SC 2000: 53C35, 53C40, 57S15 1. Introduction Recall some basic concepts on isometric actions which will be used throughout the paper. Regarding an isometric action α : L × N → N of a compact connected Lie group L on a Riemannian manifold N, a closed (totally geodesic) submanifold C is said to be a section if C intersects orthogonally all the orbits of L, and in this case α is called polar. An isometric action is said to be hyperpolar if it admits sections which are Ωat totally geodesic submanifolds. In symmetric spaces the actions of isotropy subgroups present evident examples for hyperpolar ones.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag

∗ This research was supported by the Hungarian National Science and Research Foundation OTKA T032478.