On the Volume of a Symmetric Tetrahedron in Hyperbolic and Spherical Spaces

We obtain some elementary formulas for the volume of a symmetric tetrahedron in hyperbolic and spherical spaces.     

n维双曲空间和n维球面空间中的正弦定理及应用

本文研究了n维双曲空间和n维球面空间中单形的正弦定理和相关几何不等式. 应用距离几何的理论和方法, 给出了n维双曲空间和n维球面空间中一种新形式的正弦定理, 利用建立的正弦定理获得了Hadamard 型和Veljan-Korchmaros型不等式. 另外, 建立了涉及两个n维双曲单形和n维球面单形的"度量加"的一些几何不等式.     

Spherical Harmonic Analysis for Spinors on H n (C)

Recent results on the harmonic analysis of spinor fields on the complex hyperbolic space H ⁿ (C) are reviewed. We discuss the action of the invariant differential operators on the Poisson transforms, the theory of spherical functions and the spherical transform. The inversion formula, the Paley–Wiener theorem, and the Plancherel theorem for the spherical transform are obtained by reduction to Jacobi analysis on L ²(R).     

On Decompositions of Compact Convex Sets

The present paper generalizes M. Edelstein's theorem on the indecomposability of compact convex sets in locally convex linear topological spaces to spherical and hyperbolic geometry. Moreover, the indecomposability of compact intervals in EU¹ w.r.t. homeomorphisms of EU¹ onto itself is shown.     

Thickness of a Simplex Whose Edge Lengths Fall Within a Prescribed Range

The existence of a simplex in hyperbolic, Euclidean, or spherical space whose edge lengths lie within a prescribed range has been studied in a few papers. The question considered there can be put this way. Let ? be a positive number (with a natural upper bound in the spherical case). What is the minimum number λ = λ _n (?) such that an n-simplex exists in the n-dimensional space whenever its edge lengths fall within [λ, ?]? The papers derive exact expressions for λ _n (?) in each of the three geometries. They do not tell, however, how 'thick' at least the simplex will be depending on ? and a lower bound µ ε [ λ, ?] of the edge lengths. Such a dependence clearly exists. One can easily guess for instance that, when µ is close to ?, the simplex cannot be too thin compared to the regular simplex for which µ =?. We define here a suitable thickness of a convex body and then estimate from below this thickness of a simplex in terms of n, ?, and µ for each of the three geometries.     

Decomposition of an Updated Correlation Matrix via Hyperbolic Transformations

An algorithm for hyperbolic singular value decomposition of a given complex matrix based on hyperbolic Householder and Givens transformation matrices is described in detail. The main application of this algorithm is the decomposition of an updated correlation matrix.     

Singularities of Hyperbolic Gauss Maps

In this paper we adopt the hyperboloid in Minkowski space asthe model of hyperbolic space. We define the hyperbolic Gaussmap and the hyperbolic Gauss indicatrix of a hypersurface inhyperbolic space. The hyperbolic Gauss map has been introducedby Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135]in the Poincaré ball model, which is very useful forthe study of constant mean curvature surfaces. However, it isvery hard to perform the calculation because it has an intrinsicform. Here, we give an extrinsic definition and we study thesingularities. In the study of the singularities of the hyperbolicGauss map (indicatrix), we find that the hyperbolic Gauss indicatrixis much easier to calculate. We introduce the notion of hyperbolicGauss–Kronecker curvature whose zero sets correspond tothe singular set of the hyperbolic Gauss map (indicatrix). Wealso develop a local differential geometry of hypersurfacesconcerning their contact with hyperhorospheres. 2000 MathematicalSubject Classification: 53A25, 53A05, 58C27.     

双曲矩阵分解修正问题的一些结果

This paper considers the updating problem of the hyperbolic matrix factorizations. The sufficient conditions for the existence of the updated hyperbolic matrix factorizations are first provided. Then, some differential inequalities and first order perturbation expansions for the updated hyperbolic factors are derived. These results generalize the corresponding ones for the updating problem of the classical QR factorization obtained by Jiguang SUN.     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics

A hyperbolic 3-manifold is said to have the spd-property if all its closed geodesics are simple and pairwise disjoint. For a 3-manifold which supports a geometrically finite hyperbolic structure we show the following dichotomy: either the generic hyperbolic structure has the spd-property or no hyperbolic structure has the spd-property. Both cases are shown to occur. In particular, we prove that the generic hyperbolic structure on the interior of a handlebody (or a surface cross an interval) of negative Euler characteristic has the spd-property. Simplicity and disjointness are consequences of a variational result for hyperbolic surfaces. Namely, the intersection angle between closed geodesics varies nontrivially under deformation of a hyperbolic surface.