Diameter rigidity of spherical polyhedra
The main result of the paper is the classification of the geodesically complete, compact, 2-dimensional spherical polyhedra of diameter and injectivity radius \(\pi\). It turns out that any such polyhedron is either (1) a thick spherical building of type \(A_3\) or \(B_3\), or (2) a spherical join of a finite set with a metric graph of injectivity radius greater than or equal to \(\pi\), or (3) a hemispherex. (A spherical complex is called ``hemispherex'' if it is obtained from the unit sphere by attaching unit hemispheres along great hyperspheres so that no pair of antipodal points on the sphere belongs to all hyperspheres). This result is motivated by the rank rigidity problem for spaces of nonpositive curvature which asks for a classification of such complete, simply connected spaces of rank at least two having a co-compact group of isometries. 2-dimensional spherical polyhedra of diameter and injectivity radius \(\pi\) arise as links of nonpositively curved Euclidean polyhedra having the rank greater than or equal to 2. Thus their classification is an important step towards the solution of the rank rigidity problem for 3-dimensional nonpositively curved Euclidean polyhedra. The paper contains a lot of geometric information on 2-dimensional spherical polyhedra of diameter and injectivity radius \(\pi\). The main result has recently been used for the solution of the 3-dimensional rank rigidity problem obtained by the authors in [Am. J. Math. 122, 873-885 (2000; Zbl 1041.53029)].
scientific article; zbMATH DE number 1425203