Noncompact Symmetric Spaces Of Rank Research Articles

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5), and textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1.

Canonical Extension of Submanifolds and Foliations in Noncompact Symmetric Spaces

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several applications yields the first examples of inhomogeneous isoparametric hypersurfaces in noncompact symmetric spaces of rank at least two.

Solution of a Busemann problem

We give a solution to the problem posed by Busemann which is as follows: Determine the noncompact Busemann G-spaces such that for every two geodesics there exists a motion taking one to the other. We prove that each of these spaces is isometric to the Euclidean space or to one of the noncompact symmetric spaces of rank 1 (of negative sectional curvature).

Approximation by K-finite functions in L p spaces

Let Γ ⊂ ℝn, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in Lp(ℝn) for \(p \geqslant \tfrac{{2n}}{{n + 1}}\). A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators.

Entropy of Sobolev embeddings of radial functions and radial eigenvalues of Schrödinger operators on isotropic manifolds

AbstractWe consider Sobolev embeddings between Sobolev and Besov spaces of radial functions on noncompact symmetric spaces of rank one. An asymptotic behaviour of entropy numbers of the compact embeddings is described. The estimates are used for investigation of the negative spectrum of Schrödinger type operators. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces C H n \mathbb C H^n , n ≥ 3 n \geq 3 . For the quaternionic hyperbolic spaces H H n \mathbb H H^n , n ≥ 3 n \geq 3 , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

Approximation numbers of Sobolev embeddings of radial functions on isotropic manifolds

We regard the compact Sobolev embeddings between Besov and Sobolev spaces of radial functions on noncompact symmetric spaces of rank one. The asymptotic formula for the behaviour of approximation numbers of these embeddings is described.

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type.

Some solvable stochastic control problems in noncompact symmetric spaces of rank one

A family of stochastic control problems for diffusion processes in noncompact symmetric spaces of rank one are described and explicitly solved. The problem is to control Brownian motion in a noncompact symmetric space of rank one by a drift so that it remains close to a fixed point in the space called the origin. The solution is obtained by finding a smooth solution to the Hamilton-Jaobbi or dynamic programming equation for the control problem. All noncompact symmetric spaces of rank one and dimension ≥2 are included and for each of these symmetric spaces a countably infinite family of different, explicitly solvable control problems are given. The cost functional for these control problems are obtained from some spherical functions on these spaces. These spherical functions are monotone increasing functions of the distance of a point from the origin.

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).