Special isoparametric orbits in Riemannian symmetric spaces

Geometriae Dedicata (Impact Factor: 0.47). 04/1995; 55(3):305-317. DOI: 10.1007/BF01266321

ABSTRACT In the present paper orbits of isotropy subgroups in Riemannian symmetric spaces are discussed. Principal orbits of an isotropy subgroup are isoparametric in the sense of Palais and Terng (seeCritical Point Theory and Submanifold Geometry, Springer-Verlag, Berlin, 1988). We show that excepting some special cases, the shape operator with respect to the radial unit vector field determines a totally geodesic foliation on a given principal orbit. Furthermore, we prove that the shape operators and the curvature endomorphisms with respect to the normal vectors commute on these isoparametric submanifolds.