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.
[Show abstract][Hide abstract] 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.
[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.
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