Diameter rigidity of spherical polyhedra
Diameter rigidity of spherical polyhedra
复制标题
球面多面体的直径刚度
DOI：
10.1215/s0012-7094-99-09711-9
复制
发表时间：
1999
影响因子：
2.5
通讯作者：
M. Brin
中科院分区：
文献类型：
--
作者：
W. Ballmann;M. Brin
1. 介绍。本文考虑的直径刚度问题是由非正曲率空间的秩刚度问题派生而来的。如果Y中的每个测地线段都包含在等距嵌入的凸欧几里得平面中，则非正曲率的完全单连通空间Y的秩大于或等于2。秩刚性问题要求对这样的空间进行分类，至少当Y的等距群很大时。回想一下，如果拓扑空间允许三角剖分，则称为多面体。一个具有度量长度的多面体，如果它允许三角剖分成欧几里得（分别为球面）简单体，则称为欧几里得（分别为球面）多面体。这里，欧几里得（分别为球面）k-单纯形是这样的k-单纯形a，即具有诱导长度度规的a在一般位置上与k + 1个封闭半空间（分别为sk中的封闭半球）的交点等距。研究秩大于等于2的非正曲率单连通欧几里得多面体的秩刚性问题。我们希望它们是欧几里德式的建筑或产品。dim Y = 2时的阶刚度不难求，见[BB， Section 6]。对于欧几里得多面体Y， p∈Y，我们用SpY表示Y在p处的连杆，即p处的方向集。显然，SpY是一个球面多面体，如果Y具有非正曲率，则SpY的注入半径为π。更进一步，如果Y的秩大于等于2，则SpY是测地线完备的，其直径为π。因此，对于dim Y = 3，链路X = SpY是一个测地线完备、紧致、直径和注入半径为π的二维球面多面体。本文的目的是对这样的空间x进行分类。这里有一些直径和注入半径为π的测地线完全紧致球多面体的例子。1.1. 球形的大楼。如果X是一个球形建筑，那么X带有一个自然度量，其中公寓是单位球体。对于这个度规，X的直径和注入半径为π。如果fY是具有自然度量的n≥2维欧几里德建筑，则Y中的每个测地线都包含在等距嵌入的凸欧几里德n空间中。Y中一个顶点的连杆是一个维数为n−1的球形建筑物，其注入半径和直径为π。一个n维建筑X
1. Introduction. The diameter rigidity question considered in this paper is motivated by the rank rigidity problem for spaces of nonpositive curvature. The rank of a complete, simply connected space Y of nonpositive curvature is greater than or equal to 2 if every geodesic segment in Y is contained in an isometrically embedded, convex Euclidean plane. The rank rigidity problem asks for a classification of such spaces, at least when the isometry group of Y is large. Recall that a topological space is called a polyhedron if it admits a triangulation. A polyhedron with a length metric is called Euclidean (respectively, spherical) if it admits a triangulation into Euclidean (respectively, spherical) simplices. Here a Euclidean (respectively, spherical) k-simplex is a k-simplex A such that A with the induced length metric is isometric to the intersection of k + 1 closed half-spaces in R k (respectively, closed hemispheres in S k ) in general position. We are interested in the rank rigidity of simply connected Euclidean polyhedra of nonpositive curvature and of rank greater than or equal to 2. We expect them to be Euclidean buildings or products. The rank rigidity for dim Y = 2 is not difficult and is contained in [BB, Section 6]. For a Euclidean polyhedron Y and p ∈ Y , we denote by SpY the link of Y at p, that is, the set of directions at p. Clearly, SpY is a spherical polyhedron, and, if Y has nonpositive curvature, then the injectivity radius of SpY is π . Furthermore, if the rank of Y is greater than or equal to 2, then SpY is geodesically complete and has diameter π . Hence, for dim Y = 3 the link X = SpY is a geodesically complete, compact, 2-dimensional spherical polyhedron of diameter and injectivity radius π . The aim of this paper is to classify such spaces X. Here are examples of geodesically complete compact spherical polyhedra of diameter and injectivity radius π . 1.1. Spherical building. If X is a spherical building, then X carries a natural metric, for which the apartments are unit spheres. For this metric, the diameter and injectivity radius of X are π .I fY is a Euclidean building of dimension n ≥ 2 with the natural metric, then every geodesic in Y is contained in an isometrically embedded, convex Euclidean n-space. The link of a vertex in Y is a spherical building of dimension n−1, which has injectivity radius and diameter π .A nn-dimensional building X