Article

# Special isoparametric orbits in Riemannian symmetric spaces

Geometriae Dedicata (Impact Factor: 0.52). 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.

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**ABSTRACT:**We characterize the symmetric space ＄M=Sp(n)/U(n)＄ by using the shape operator of small geodesic spheres in ＄M＄ , and a certain tensor field that satisfies various algebraic properties. We also give a partial generalization to any isotropy irreducible symmetric space.01/2006; - [Show abstract] [Hide abstract]

**ABSTRACT:**Given a compact symmetric space, M, we obtain the mean exit time function from a principal orbit, for a Brownian particle starting and moving in a generalized ball whose boundary is the principal orbit. We also obtain the mean exit time function of a tube of radius r around special totally geodesic submanifolds P of M. Finally we give a comparison result for the mean exit time function of tubes around submanifolds in Riemannian manifolds, using these totally geodesic submanifolds in compact symmetric spaces as a model.Acta Mathematica Sinica 05/2005; 21(3):555-562. DOI:10.1007/s10114-004-0479-z · 0.48 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Let Pt denote the tubular hypersurface of radius t around a given compatible submanifold in a symmetric space of arbitrary rank. The authors will obtain some relations between the integrated mean curvatures of Pt and their derivatives with respect to t. Moreover, the authors will emphasize the differences between the results obtained for rank one and arbitrary rank symmetric spaces.Chinese Annals of Mathematics 04/2012; 23(01). DOI:10.1142/S0252959902000067 · 0.45 Impact Factor

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