Abstract. We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.
A geometric formulation of the generalized Law of Sines for simplices in constant curvature spaces is presented. It is explained how the Law of Sines can be seen as an instance of the so-called polar duality, which can be formulated as a duality between Gram matrices representing the simplex. (2000): 51M20, 52A55, 52B11.
Mathematics Subject Classification
Abstract. We discuss and compare the notions of ideal boundaries, Floyd boundaries and Gromov boundaries of metric spaces. The three types of boundaries at infinity are compared in the general setting of unbounded length spaces as well as in the special cases of CAT(0) and Gromov hyperbolic spaces. Gromov boundaries, usually defined only for Gromov hyperbolic spaces, are extended to arbitrary metric spaces.
IntroductionThere are various notions of "boundaries at infinity" of metric spaces in the literature. Perhaps the best known is the ideal boundary ∂ I X defined using geodesic rays, and particularly studied for the classes of CAT(0) and proper geodesic Gromov hyperbolic spaces. Closely related to this is the concept of a Gromov boundary ∂ G X defined using Gromov sequences (or "sequences converging to infinity"). For more on both of these concepts, see for instance [BH], [GH], [CDP], [BHK], and [V]. The Gromov boundary is usually defined only for Gromov hyperbolic spaces, but we extend this concept to arbitrary metric spaces.A third type of boundary at infinity is the g-boundary ∂ g X of an unbounded length space which replaces an unbounded metric l by a bounded metric σ obtained via a conformal distortion involving a suitable function g. The spherical and Floyd boundaries are both just the g-boundary, but with g restricted to lie in certain nice classes of functions; many results involving these concepts are independent of the choice of g. The spherical boundary arises as a byproduct of sphericalization, a concept introduced in The current paper aims to shed more light on ∂ g X by comparing and contrasting it with the ideal and Gromov boundaries. Understanding the spherical boundary
We develop the utility of Gram matrix machinery as a tool to treat the geometry of simplices in space forms. A formula relating the determinant of a normalized Gram matrix to the geometry of the simplex it represents is presented. We then apply the tools to leaf spaces, i.e. the set of degree 1 vertices of a metric tree. One main result is that for a given metric space X there exists a constant κ 0 < 0, such that X embeds into all hyperbolic spaces of curvature less than κ 0 , if and only if X is a leaf space.
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