Symmetric Spaces Of Rank Research Articles

Linking integrals on rank-1-symmetric spaces

In symmetric spaces of rank one we obtain a right inverse of the Cartan differential on exact differential forms as an integral operator. As a corollary we extend Gauss’ linking integral beyond space forms.

Full Laplace spectrum of distance spheres in symmetric spaces of rank one

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.

Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

Abstract We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

Hyperbolic rank rigidity for manifolds of -pinched negative curvature

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

A Dehn function for Sp(2n, ℤ)

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least [Formula: see text] should have at most polynomial Dehn function. We prove that the lattice [Formula: see text] has quadratic Dehn function when [Formula: see text]. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

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.

Positively curved Riemannian metrics with logarithmic symmetry rank bounds

We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.

The spectrum and heat dynamics of locally symmetric spaces of higher rank

The aim of this paper is to study the spectrum of the$L^{p}$Laplacian and the dynamics of the$L^{p}$heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank-one case, it turns out that the$L^{p}$heat semigroup on$M$has a certain chaotic behavior if$p\in (1,2)$, whereas for$p\geq 2$such chaotic behavior never occurs.

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.

An introduction to the theory of generalized conics and their applications

A generalized conic is a set of points with the same average distance from the pointset Γ in the Euclidean coordinate space. The measuring of the average distance is realized via integration over Γ as the set of foci. Using generalized conics we give a process for constructing convex bodies which are invariant under a fixed subgroup G of the orthogonal group in R n . The motivation is to present the existence of non-Euclidean Minkowski functionals with G ⊂ O ( n ) in the linear isometry group provided that the closure of G is not transitive on the unit sphere. As an application, consider R n as the tangent space at a point of a connected Riemannian manifold M and G as the holonomy group. If the holonomy group is not transitive on the unit sphere in the tangent space, then the Lévi-Civita connection is (re)metrizable in the sense that there is a smooth collection of non-Euclidean Minkowski functionals on the tangent spaces such that it is invariant under parallel transport with respect to the Lévi-Civita connection (according to Berger’s list of possible Riemannian holonomy groups, all of them are transitive on the unit sphere in the tangent space except in the case where the manifold is a symmetric space of rank≥2). We present the (re)metrizability theorem in a more general context of metrical linear connections with a torsion tensor that is not necessarily vanishing. This allows us to declare eight classes of manifolds equipped with an invariant smooth collection of Minkowski functionals on the tangent spaces. They are called Berwald manifolds in a general sense.

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).

On spectral expansions of piecewise smooth functions depending on the geodesic distance

We consider the expansion of a piecewise smooth function depending on the geodesic distance to some point in the eigenfunctions of the Beltrami-Laplace operator on an n-dimensional symmetric space of rank 1. We show that if the expansion converges at this point, then the function must have continuous derivatives up to and including the order (n − 3)/2.

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n − 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that \({{\rm{tr}}(R_{v}^{k})}\), v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n − 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n − 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.

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.

Killing helices on a symmetric space of rank one

In this paper, we study helices which are orbits of one parameter families of isometries on a symmetric space of rank one. We introduce the notion of structure torsion fields for helices, show the necessary and sufficient condition that they are generated by some Killing vector fields, and study their moduli space.

Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly

The global multiplicative properties of Laplace type operators acting on irreducible rank one symmetric spaces are considered. The explicit form of the multiplicative anomaly is derived and its corresponding value is calculated exactly, for important classes of locally symmetric spaces and different dimensions.

Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one

We show some integral representations of the heat kernels and explicit expressions of the Green functions for the Laplace–Beltrami operators on three series of hyperbolic spaces.

Some Properties of Boundaries of Symmetric Spaces of Rank One

M. Cowling, A. H. Dooley, A. Koranyi and F. Ricci used groups of Heisenberg type in order to study the symmetric spaces of rank one of noncompact type in a unified manner. This paper extends this work by using the same formulation to investigate the boundaries of these spaces. In particular, we prove a conjecture of Koranyi concerning metrics on the boundary and demonstrate that the classical Cayley transform extends to a 1-quasiconformal map of the boundary.

A Formula for the First Regularized Trace of a Perturbed Laplace–Beltrami Operator

The theory of regularized traces is well developed for problems induced by ordinary differential expressions. It was shown in the fundamental paper [1] that the derivation of trace formulas can be reduced to studying zeros of entire functions possessing a specific asymptotic structure related to the particular form of the fundamental solution system of the differential equation. The case of problems induced by partial differential operators is much harder. This is primarily due to the complicated structure of the spectrum. The asymptotic spectrum distribution was obtained in [2] for the operator −∆+V with smooth odd potential V on the sphere S. However, the results of [2] do not allow finding regularized trace formulas. Further substantial progress in this problem was made in [3], where a formula for the first regularized trace was obtained for the first time for the operator −∆ + V with odd smooth potential V . Later, the method of [4] was used in [5] for finding the first regularized trace for an odd potential with n = 2 and in [6] for deriving a regularized trace formula for the operator −∆+V on compact symmetric spaces of rank 1. However, the condition that the potential V is C∞ was substantially used in all these papers. In the present paper, we derive a formula for the first trace of the operator −∆ +V on S in the case of a finite smoothness of the potential V and without the assumption that V is odd. If the potential is odd, then formula (1) coincides with the corresponding formula in [5]. Let L0 = −∆, and let V be the operator of multiplication by a real smooth function in L2 (S), where S is the two-dimensional sphere. By L we denote the operator L0 +V . Further, let R0(z) = (L0 − zE) and R(z) = (L− zE)−1; let λn = n(n+ 1) and μ n , n = 0, 1, . . . , −n ≤ i ≤ n, be the eigenvalues of the operators L0 and L, respectively, and let f (i) n = Y (i) n (φ, θ) be the orthonormal spherical functions, that is, the eigenfunctions of the operator L0. The following assertion is the main result of the present paper.

The Moduli Space of Riemann Surfaces is Kahler Hyperbolic

Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant curvature -1 in the Teichmuller (=Kobayashi) metric. When $r=\dim_{\cx} \cM_{g,n}$ is greater than one, however, $\cM_{g,n}$ carries no complete metric of bounded negative curvature. Instead, Dehn twists give chains of subgroups $\zed^r \subset \pi_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$. In this paper we introduce a new Kahler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank.