Geometry and Combinatorics

The subject of this Ph.D.-thesis is somewhere in between continuous and discrete geometry. Chapter 2 treats the geometry of finite point sets in semi-Riemannian hyperquadrics,using a matrix whose entries are a trigonometric function of relative distances in a given point set. The distance introduced on the semi-Riemannian space forms has complex values and is an extension of the usual Riemannian distance on the simply connected space forms. One of the most important results of the chapter is Theorem 2, that relates the determinant of the previously mentioned trigonometric matrix to the geometry of a simplex in a semi-Riemannian hyperquadric. In chapter 3 we study which finite metric spaces that are realizable in a hyperbolic space in the limit where curvature goes to -∞. We show that such spaces are the so called leaf spaces, the set of degree 1 vertices of weighted trees. We also establish results on the limiting geometry of such an isometrically realized leaf space simplex in hyperbolic space, when curvature goes to -∞. Chapter 4 discusses negative type of metric spaces. We give a measure theoretic treatment of this concept and related invariants. The theory developed is then applied to show, that hyperbolic spaces are of strictly negative type. We also give an application to maximal distributions of subharmonic kernels. The most important application is probably the discussion of closed geodesics and negative type. Among other things we show, that a compact Riemannian manifold of negative type and dimension at least 2 is simply connected.