Compact Symmetric Spaces Of Rank Research Articles

Explicit computations of some spectral invariants of compact symmetric spaces of rank one

In this paper, we propose a unified approach for the study of the heat functions of the compact symmetric spaces of rank one. One of the consequences of this study is to show that the spectral properties of any compact symmetric space of rank one can be determined by a given polynomial. We prove that the coefficients of these polynomials are related to the Stirling numbers of the first kind. Moreover, we show that these polynomials possess some combinatorial interpretations, generalizing some combinatorial properties of Bernoulli numbers.

Adaptive estimation of nonparametric geometric graphs

This article studies the recovery of graphons when they are convolution kernels on compact (symmetric) metric spaces. This case is of particular interest since it covers the situation where the probability of an edge depends only on some unknown nonparametric function of the distance between latent points, referred to as Nonparametric Geometric Graphs (NGG). In this setting, almost minimax adaptive estimation of NGG is possible using a spectral procedure combined with a Goldenshluger-Lepski adaptation method. The latent spaces covered by our framework encompasses (among others) compact symmetric spaces of rank one, namely real spheres and projective spaces. For these latter, explicit computations of the eigenbasis and of the model complexity can be achieved, leading to quantitative non-asymptotic results. The time complexity of our method scales cubicly in the size of the graph and exponentially in the regularity of the graphon. Hence, this paper offers an algorithmically and theoretically efficient procedure to estimate smooth NGG. As a by product, this paper shows a non-asymptotic concentration result on the spectrum of integral operators defined by symmetric kernels (not necessarily positive).

Smoothness of the Radon–Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

Smoothness of the Radon–Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

A unified approach to compact symmetric spaces of rank one

A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.

Totally Geodesic Submanifolds in Compact Symmetric Spaces of Rank Two

In 1978 B. Y. Chen and T. Nagano obtained the local classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we investigate their global classification.

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.

Multivariate polynomial inequalities with respect to doubling weights and [formula omitted] weights

In one-dimensional case, various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz–Zygmund, Nikolskii, Schur, Remez, etc., have been proved under the doubling condition or the slightly stronger A ∞ condition on the weights by Mastroianni and Totik in a recent paper [G. Mastroianni, V. Totik, Weighted polynomial inequalities with doubling and A ∞ weights, Constr. Approx. 16 (1) (2000) 37–71]. The main purpose of this paper is to prove multivariate analogues of these results. We establish analogous weighted polynomial inequalities on some multivariate domains, such as the unit sphere S d − 1 , the unit ball B d , and the general compact symmetric spaces of rank one. Moreover, positive cubature formulae based on function values at scattered sites are established with respect to the doubling weights on these multivariate domains. Some of these multi-dimensional results are new even in the unweighted case. Our proofs are based on the investigation of a new maximal function for spherical polynomials.

The nullity of a compact minimal hypersurface in a compact symmetric space of rank one

We determine a compact minimal hypersuface with the least nullity in the Cayley projective plane. Combining this with the preceding results, we conclude the following: Let X be a compact symmetric space of rank one and M a compact minimal hypersuface in X. Then the nullity of M is bounded from below by the dimension of X. When the nullity of M is equal to the dimension of X, M must be a minimal geodesic hypersphere in X. Conversely,the nullity of a minimal geodesic hypersphere in X is equal to the dimension of X.

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.

Spectral zeta functions for compact symmetric spaces of rank one

Spectral zeta functions for compact symmetric spaces of rank one

On the Analytic Continuation of the Minakshisundaram–Pleijel Zeta Function for Compact Symmetric Spaces of Rank One

We give two equivalent analytic continuations of the Minakshisundaram–Pleijel zeta function ζU/K(z) for a Riemannian symmetric space of the compact type of rank oneU/K. First we prove that ζU/Kcan be written asζU/K(z)=eiπ(z−N/2)VU/KζG/K(z)+F(z),whereN=dimU/K,VU/Kis the volume ofU/K, ζG/K(z) is the local zeta function forG/K(the noncompact symmetric space dual toU/K), andF(z) is an analytic function which is given explicitly as a contour integral (cf. Eq. (4.11)). To prove the above formula we use a relation (first noticed by Vretare) between the scalar degeneracies of the Laplacian onU/Kand the Plancherel measure onG/K. The second expression we obtain for ζU/K(z) is in terms of a series of (generalized) Riemann zeta functions ζR(z,q) (cf. Eq. (5.9)). The doubly connected case of real projective spaces is also discussed.

Approximations on compact symmetric spaces of rank 1

On an arbitrary Riemannian symmetric space of rank 1 the Nikol'skii classes are defined by considering differences along geodesics. These spaces are described in terms of the best approximations by polynomials in spherical harmonics on , that is, by linear combinations of the eigenfunctions of the Laplace-Beltrami operator on . The results of Nikol'skii and Lizorkin on the approximation of functions on the sphere are generalized.

On lower bounds on the size of designs in compact symmetric spaces of rank 1

The concept of t-designs in compact symmetric spaces of rank 1 is a generalization of the theory of classical t-designs. In this paper we obtain new lower bounds on the cardinality of designs in projective compact symmetric spaces of rank 1. With one exception our bounds are the first improvements of the classical bounds by more than one. We use the linear programming technique and follow the approach we have proposed for spherical codes and designs. Some examples are shown and compared with the classical bounds.

Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups

In this paper we prove central limit theorems of the following kind: let $S^d \subset \mathbb{R}^{d + 1}$ be the unit sphere of dimension $d \geq 2$ with uniform distribution $\omega_d$. For each $k \epsilon \mathbb{N}$, consider the isotropic random walk $(X_n^k)_{n \geq 0}$ on $S^d$ starting at the north pole with jumps of fixed sizes $\angle (X_n^k, X_{n - 1}^k) = \pi/\sqrt{k}$ for all $n \geq 1$. Then there is some $k_0(d)$ such that for all $k \geq k_0(d)$, the distributions $\varrho_k$ of $X_k^k$ have continuous, bounded $\omega_d$-densities $f_k$. Moreover, there is a (known) Gaussian measure $\nu$ on $S^d$ with $\omega_d$-density such that $||f_k - h||_{\infty} = O(1/k)$ and $||\varrho_k - \nu|| = O(1/k)$ for $k \to \infty$, where $O(1/k)$ is sharp. We shall derive this rate of convergence in the central limit theorem more generally for a quite general class of isotropic random walks on compact symmetric spaces of rank one as well as for random walks on $[0, \pi]$ whose transition probabilities are related to product linearization formulas of Jacobi polynomials.

Riesz Means on Compact Riemannian Symmetric Spaces

AbstractWe study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K‐moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.

Isotropic Immersions

Recently, Ferus ([5], [6]) classified connected Riemannian manifolds with parallel second fundamental form in a real space form of constant curvature . In this paper we may restrict our attention to isotropic submanifolds with parallel second fundamental form in the Euclidean sphere Sm(k) of constant curvature k. Due to Ferus, we find that an isotropic submanifold with parallel second fundamental form in Sm is locally congruent to one of compact symmetric spaces of rank one and the immersion is locally equivalent to the second or the first standard immersion according as M is a sphere or not. In Section 2, we characterize the first standard immersion of a complex projective space into a sphere in terms of isotropic immersions.

On almost Blaschke manifolds II

where Br(Om) denotes the metric r-ball centered at the origin o m in the tangent space TraM and ExPm denotes the exponential mapping from T~M onto M. In general the diameter d(M) is equal to or greater than the injectivity radius i(M). In special case when the diameter is equal to the injectivity radius, we say that such (M, g) is a Blaschke manifold. For example compact symmetric spaces of rank one so called CROSSes have such a property. The famous Blaschke conjecture asks; Is any Blaschke manifold isometric to a CROSS? It is known that Blaschke manifolds have the cohomology types of the CROSSes, although in I-2, 11] R.Bott and H.Nakagawa have shown that this conclusion holds under much weaker assumption. Moreover M. Berger proved that Blaschke manifolds which are diffeomorphic to spheres or real projective spaces are isometric to standard ones ([1]). Recently it was shown that any Blaschke manifold is homeomorphic to one of the CROSSes in low dimensions by H. Gluck, F. Warner and C.T. Yang [7]. See also H. Sato and T. Mizutani [14, 15]. In this paper we study the following problem from a different view point. What is the topology of a riemannian manifold whose injectivity radius is close to its diameter? We get the following results in the lower dimensional case.

On tight $t$-designs in compact symmetric spaces of rank one

On tight $t$-designs in compact symmetric spaces of rank one

Bohr-Favard inequality for functions on compact symmetric spaces of rank one

Bohr-Favard inequality for functions on compact symmetric spaces of rank one

Antipodal Manifolds in Compact Symmetric Spaces of Rank One

Antipodal Manifolds in Compact Symmetric Spaces of Rank One