Riemannian Symmetric Spaces Of Rank Research Articles

Lipschitz and Fourier type conditions with moduli of continuity in rank 1 symmetric spaces

Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an $L^2$ function, in the setting of Riemannian symmetric spaces of rank $1$ whose growth depends on a $k$th-order modulus of continuity.

POINT DISTRIBUTIONS IN TWO‐POINT HOMOGENEOUS SPACES

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of -designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

Resonances and Scattering Poles in Symmetric Spaces of Rank One

Abstract We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of Kashiwara–Ōshima. From this, we derive that the poles of the corresponding meromorphic continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp–Stein intertwiner. As a by-product of the underlying ideas, we obtain a new and self-contained proof of Helgason’s conjecture for distributions in the case of rank one symmetric spaces.

Rigidity of negatively curved geodesic orbit Finsler spaces

We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic orbit Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two

We study special Lagrangian submanifolds of the cotangent bundle $T^*S^n$ of the sphere in the tangent space of Riemannian symmetric space of rank two. We show that the special Lagrangian submanifolds correspond to the solution of a differential equation on $\mathbb{R}^2$ under the assumption that the submanifold is of cohomogeneity one. Our result is the generalization of the former work of Sakai and the first author [5]. We study the qualitative properties of the solution for the special Lagrangian submanifolds and give some examples.

Biharmonic hypersurfaces in Riemannian symmetric spaces I

We classify biharmonic geodesic spheres in the Cayley projective plane. Our results completes the classification of all biharmonic homogeneous hypersurfaces in simply connected compact Riemannian symmetric spaces of rank 1. In addition we show that complex Grassmannian manifolds, and exceptional Lie groups $F_4$ and $G_2$ admit proper biharmonic real hypersurfaces.

Some remarks on polar actions

We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.

Riesz Transforms on Compact Riemannian Symmetric Spaces of Rank One

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi–Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia’s extrapolation theorem. The latter results are of independent interest.

COHOMOGENEITY ONE ACTIONS ON SOME NONCOMPACT SYMMETRIC SPACES OF RANK TWO

We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact Riemannian symmetric spaces of rank two G ℂ2 /G2, SL3(ℂ)/SU3 and SO 02,n + 2 /SO2SO n + 2, n ≥ 1.

Fractional integrals on compact Riemannian symmetric spaces of rank one

In this paper we study mixed norm boundedness for fractional integrals related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU 2, m/S(U2Um) of noncompact type.

Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one

We study the problem of inversion of the local Pompeiu transform on Riemannian symmetric spaces of rank one. The explicit formula for the reconstruction of a function by its averages over balls and spheres with a single fixed radius is obtained.

On a problem of Berenstein-Gay and its generalizations

We obtain a solution of the Berenstein-Gay problem on the local analogue of spectral analysis on Riemannian symmetric spaces of rank 1. The proof is based on constructing transmutation maps connected with eigenfunction expansions of the Laplace-Beltrami operator on .

The role of restriction theorems in harmonic analysis on harmonic NA groups

We shall obtain inequalities for Fourier transform via moduli of continuity on NA groups. These results in particular settle the conjecture posed in a recent paper by W.O. Bray and M. Pinsky in the context of noncompact rank one symmetric spaces. These problems naturally demand versions of Fourier restriction theorem on these spaces which we shall prove. We shall also elaborate on the connection between the restriction theorem and the Kunze–Stein phenomena on NA groups. For noncompact Riemannian symmetric spaces of rank one analogues of all the results follow the same way.

On the Schwartz space isomorphism theorem for rank one symmetric space

In this paper we give a simpler proof of the L p-Schwartz space isomorphism (0 < p ≤ 2) under the Fourier transform for the class of functions of left δ-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s [2] proof of the corresponding result in the case of left K-invariant functions on X. Thus we give a proof which relies only on the Paley-Wiener theorem.

A generalization of the Cartan–Helgason theorem for Riemannian symmetric spaces of rank one

Let U / K be a compact Riemannian symmetric space with U simply connected and K connected. Let G / K be the noncompact dual space, with G and U analytic subgroups of the simply connected complexification GC. Let G = KAN be an Iwasawa decomposition of G, and let M be the centralizer of A in K. For d in U, let µ be the highest restricted weight of d, and let s be the M-type acting in the highest restricted weight subspace of Hd. Fix a K-type t. Earlier we proved that if U / K has rank one, then d|K contains t if and only if t|M contains s and µ in µs,t + ?sph, where ?sph is the set of highest restricted spherical weights and µs,t is a suitable element of a* uniquely determined by s and t. In this paper we obtain an explicit formula for this element in the case of U / K = Sn, Pn(C), Pn(H). This gives a generalization of the Cartan?Helgason theorem to arbitrary K-types on these rank one symmetric spaces.

Exact and semiclassical approach to a class of singular integral operators arising in fluid mechanics and quantum field theory

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated to Riemannian symmetric spaces of rank one with positive, negative or vanishing curvature. For all other cases an accurate semiclassical approximation is derived, based on the identification of the operators with a peculiar Schroedinger-like operator.

Evidence of predictive smooth pursuit eye movement deficits in schizophrenia using a newly developed cued target paradigm

Let M be a compact Riemannian symmetric space. We give an analytical expression for the area and volume functions of geodesic balls in M and for the area and volume functions of tubes around some totally geodesic submanifolds P of M. We plot the graphs of these functions for some compact irreducible Riemannian symmetric spaces of rank two.

Smooth pursuit eye movement abnormalities in children with and vulnerable to schizophrenia

Let M be a compact Riemannian symmetric space. We give an analytical expression for the area and volume functions of geodesic balls in M and for the area and volume functions of tubes around some totally geodesic submanifolds P of M. We plot the graphs of these functions for some compact irreducible Riemannian symmetric spaces of rank two.

Correlates of mismatch negativity reduction in schizophrenia

Let M be a compact Riemannian symmetric space. We give an analytical expression for the area and volume functions of geodesic balls in M and for the area and volume functions of tubes around some totally geodesic submanifolds P of M. We plot the graphs of these functions for some compact irreducible Riemannian symmetric spaces of rank two.