到欧氏空间的等距极小浸入

本文研究了欧氏空间极小子流形的测地球的体积增长,给出了一个黎曼流形可等距极小浸入到欧氏空间的一个必要条件,并给出了具非正截曲率的欧氏空间极小超曲面的一个分类．     

欧氏空间中闭子流形的拓扑

本文通过建立浸入在欧氏空间或欧氏球面中某些子流形的Ricci曲率的下界估计和同调群的消失定理,确定了这些子流形的拓扑特征。     

欧氏空间中闭子流形的拓扑

本文通过建立浸入在欧氏空间或欧氏球面中某些子流形的 Ricci 曲率的下界估计和同调群的消失定理,确定了这些子流形的拓扑特征.     

欧氏空间中子流形的曲率与几何性质

研究了欧氏空间En p中具常数量曲率n(n-1)r的n(n>2)维完备连通子流形,得到了En p截面曲率非负且法联络平坦的完备连通子流形的一个分类定理.     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

三维欧氏空间的仿射乘积曲面

定义了三维欧氏空间中的仿射乘积曲面,给出了极小仿射乘积曲面以及高斯曲率为零的仿射乘积曲面的分类,同时还给出了平均曲率为非零常数的仿射乘积曲面的分类.     

欧氏空间中子流形的Pinching现象

本文研究了欧氏空间中紧致子流形的Ｐｉｎｃｈｉｎｇ现象,得到了一些公式,并证明了一些几何量的Ｐｉｎｃｈｉｎｇ定理     

李奇曲率平行的黎曼流形到欧氏空间的等距浸入

设ｆ：Ｍｎ→Ｒｎ＋ｐ为具平行李奇曲率的黎曼流形到欧氏空间的等距浸入．对ｐ＝１，本文给出了极小条件下以及平均曲率处处非零条件下该浸入的分类     

欧氏空间中隐没子流形的中曲率向量场

在[4]中导出了隐没在欧氏空间R~(m+p)中的紧致、有向的m维子流形M~m的Minkowski公式其中K_(2r)是黎曼流形M~m的Killing不变量,x是子流形M~m(?)R~(m+p)的定位向量,H_(2r)是第r个中曲率向量场。特别是,H_0正是通常的中曲率向量场。     

欧氏空间子流形的第一特征值的估计

本文利用浸入在欧氏空间中的子流形的第二基本形式的长度平方估计其Ｌａｐｌａｃｅ算子的第一特征值的上界,从而建立紧致子流形等距同构于球面的一个特征．     

Relationship Between Quantum and Poisson Structures of Odd Dimensional Euclidean Spaces

It is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are homeomorphic to the Poisson prime and Poisson primitive spectra of the multiparameter Poisson algebra of odd-dimensional euclidean spaces in the case when the multiplicative subgroup of a base field generated by the parameters is torsion free. As a corollary, it is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring.     

Euclidean conformally flat submanifolds in codimension two obtained as intersections

We characterize a large family of codimension two euclidean conformally flat submanifolds in the class of isometrically rigid ones and construct explicit examples.

     

Euclidean matching problems and the metropolis algorithm

The Euclidean matching problem consists in partitionning a cloud ofn points inton/2 pairs such that the sum of the euclidean distances between the endpoints of each couple is minimized, resp. maximized.In this paper the thermodynamically inspired approach of simulated annealing is applied to find good approximations to large scale problems of the above kind. Also the asymptotic behaviour of random euclidean matching problems is studied, in particular computational results on the rate of convergence towards the asymptote are given.

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Zusammenfassung Euklidische Matching-Probleme bestehen darin, eine ausn Punkten bestehende Wolke derart inn/2 Paare zu partitionieren, daß die Summe der Abstände zwischen den Endpunkten der Paare minimal bzw. maximal wird. Vorliegende Arbeit beschreibt die Anwendung der simulierten Abkühlung, eines durch die Thermodynamik inspirierten Verfahrens, zur Lösung großer derartiger Probleme. Zudem wird das asymptotische Verhalten zufälliger euklidischer Matching-Probleme studiert, insbesondere werden numerische Angaben zur Konvergenzrate gegen die Asymptote gegeben.

     

On a Note From Oliver Concerning Generalized Euclidean Group Rings

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E ₂(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.     

On the multiple points of immersions in Euclidean spaces

Given a self-transverse immersion of a closed, oriented manifold in a euclidean space and a natural number we compute the oriented cobordism class of the manifold of -tuple points.

     

A Sampling Theory for Compact Sets in Euclidean Space

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points, and critical values of distance functions. The authors were partially supported by DARPA under grant HR0011-05-1-0007 and by ANR under grant “Topologie, Géométrie Différentielle et Algorithmes.”     

Conformal Metrics on the Unit Ball in Euclidean Space

We study densities on the unit ball in euclidean space whichsatisfy a Harnack type inequality and a volume growth conditionfor the measure associated with . For these densities a geometrictheory can be developed which captures many features of thetheory of quasiconformal mappings. For example, we prove generalizationsof the Gehring-Hayman theorem, the radial limit theorem andfind analogues of compression and expansion phenomena on theboundary. 1991 Mathematics Subject Classification: 30C65.     

Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.

We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

     

An equilateral four point property which characterizes generalized euclidean spaces

Various characterizations of generalized euclidean spaces among complete metric spaces which contain a metric line joining each two points make use of euclidean four point properties, which require that every quadruple from a suitably chosen class of quadruples of points of the metric space be isometric with a quadruple of euclidean points. The present paper shows if every quadruple which contains an equilateral triple and a linear triple is embeddable, the space is generalized euclidean.     

A new class of four-point properties which characterize euclidean spaces

Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.