n维双曲空间和n维球面空间中的正弦定理及应用(英文)

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.     

关于双曲空间中n维Pedoe不等式及应用

本文应用距离几何的理论与方法,研究了双曲空间中n维单形的几何不等式问题,建立了双曲空间中涉及两个单形棱长的n维Pedoe不等式和彭-常不等式,由此获得双曲空间n维单形的一些新的不等式.     

高维非Euclid几何的几个基本不等式

通过提出n维双曲空间Hⁿ中有限点集 Σ_n(H(A))共超球的概念和n维球面空间Sⁿ 中有限点集Σ_n(S(A))共超平面的概念,使得n维双曲空间Hⁿ(或球面空间Sⁿ)中共超球(或共超平面)的有限点集Σ_n(H(A))(或Σ_n(S(A)))的Cayley-Menger矩阵 (或的秩不超过n+2. 再利用特征根的方法,建立了n维双曲空间和球面空间中的杨-张型不等式、Neuberg-Pedoe型不等式以及度量加型不等式,这些几何不等式分别是n维双曲空间和球面空间中的基本不等式.另外,也提出了与此相关的一些问题和猜想.     

关于单形空间角的准正弦概念及应用

建立了欧氏空间Ｅ^ｎ中ｎ维单形空间角的准正弦概念，并应用于高维正弦定理的改进、Ｓｔｅｉｎｅｒ定理的高维推广及切点不等式的简化证明；又推出了有关单形的一些新不等式．     

关于常曲率空间中有限点集的几何不等式

本文用代数方法建立了ｎ维球面型空间Ｓ_ｎ（Ｋ）和ｎ维双曲空间Ｈ_ｎ（Ｋ）中有限点集的点与点两两之间之距离的一类几何不等式，本文还建立了ｎ维欧氏空间Ｅ^ｎ中共球有限点集的一类几何不等式．作为本文结果的应用，简洁地得出［３］中的一个重要结果，并得出Ｅ^ｎ中有限点集的两个几何不等式     

球面空间中的Pedoe不等式与张-杨不等式及应用

本文应用距离几何的理论与方法,研究了n维球面空间中n维单形与有限点集的几何不等式问题,建立了球面空间中n维单形一种形式的Pedoe不等式与有限点集一种形式的张-杨不等式,并应用它获得n维球面空间中Veljan-Korchmaros型不等式与Finsler-Hadwinger型不等式.     

双曲空间中Veljan-Korchmaros型不等式及应用

本文研究了双曲空间Hn(K)中n维高维单形的几何不等式问题.利用距离几何的理论与方法,获得了涉及n维双曲单形体积,侧面积与棱长的几个几何不等式,这些几何不等式是双曲单形几何不等式的基础.     

Bloch型空间的一个刻画

本文研究了Bloch型空间中函数性质问题.利用拟双曲度量及一些不等式得到了Bloch型空间B^α(B_n) (0< α ≤ 1)的一个新的刻画,该刻画将Bloch型空间B^α(B_n)的Holland-Walsh刻画推广到一个高阶形式.     

双曲空间H_n(-1)和球面空间S_n(1)中单形顶点角的一些不等式

对双曲空间H_n(-1)和球面空间S_n(1)中的n维单形Ω_n,建立了一些涉及k级顶点角的不等式,应用这些不等式可以获得双曲空间H_n(-1)和球面空间S_n(1)中平行体体积的不等式.     

SL(n+1,R)/S(GL(1,R)×GL(n,R))上线丛的一个超几何方程

本文研究了伪黎曼对称空间SL（n+1,R）/S（GL（1,R）×GL（n,R））线丛上的微分方程.利用李代数方法,即Casimir算子得到这个微分算子.这个微分算子是一个超几何方程,这个结论推广了文献[1,3,5]中的微分方程.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Some geometric inequalities in non-euclidean space

In this paper, we obtained three geometric inequalities for theu-dimensional polar sines and the dihedral angles of ann-dimensional simples. Besides, we obtained an inequality for the dihedral angles of ann-dimensional simplex in then-dimensional hyperbolic spaceH _n.Project Supported by National Natural Foundation P. R. China     

双曲空间形式中具有常平均曲率的超曲面(英文)

本文研究了双曲空间形式H~(n+1)(—1)中具有常平均曲率及两个离散主曲率(其中一个主曲率是1-重)的完备连通可定向的n-维超曲面M~n.利用活动标架,得到如果M~n的基本形式的模长满足刚性条件(1.3),那么M~n同构双曲柱面.     

Regular simplices and lower volume bounds for hyperbolicn-manifolds

For an-dimensional compact hyperbolic manifoldM ⁿ a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofM ⁿ. Its proof relies upon local density bounds for hyperbolic sphere packings.     

Edge Matrix of Hyperbolic Simplices

In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277–282) and a new method, we give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a spherical n-simplex.Mathematics Subject Classifications (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.     

A geometric inequality for a finite set of points on a sphere and its applications

In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S _n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ⁿ.     

On the Existence of Completely Saturated Packings and Completely Reduced Coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W.Kuperberg: every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every >0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than but which also admits a tiling.     

Compact Homogeneous Hypersurfaces in Hyperbolic Space

In this paper we prove that an n-dimensional compact homogeneous Riemannian manifold isometrically immersed in the hyperbolic space Hⁿ⁺¹ is isometric to a sphere.AMS Subject Classification (1990): 53A07, 53C42, 57S15