It is well known that the unit ball endowed with the Kobayashi metric is isometric to complex hyperbolic space and in particular is an example of a negatively curved Riemannian manifold. One would then suspect that when Ω ⊂Cd is a domain close to the unit ball then (Ω ,KΩ ) should be negatively curved (in some sense). Unfortunately, for general domains the Kobayashi metric is no longer Riemanni...

We prove that the Teichmuller space of genus g > 1 surfaces with the Teichmuller metric is not a Gromov hyperbolic space.

We show that a cocompact hyperbolic group in dimension 3 is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension 3.

Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmüller space Teich(S) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the Weil-Petersson metric has higher rank in the sense...

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmüller space. We prove that the identity map from the Teichmüller space equipped with the usual Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Furthermore, we...

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used to produce twoor three-dimensional visualizations of datasets consisting of multidimensional objects or interobject distances. Recently, metric MDS has been applied to the problems of graph embedding for the purpose of approximate encoding of edge or path costs using node coordinates in metric space. Several a...

A geodesic metric space is δ-hyperbolic in the sense of Gromov if and only if the intersection of any two metric balls is almost a ball. In particular, R-trees can be characterised by the property that the intersection of any two metric balls is again a metric ball.

1. Introduction. The unit disc furnished with the Poincaré metric provides a model (the Poincaré model) for the hyperbolic geometry. The linear fractional transformations of the unit disc onto itself constitute the group of motions. An analogous phenomenon arises in the unit ball of any complex Hubert space when it is furnished with the Carathéodory-Reiffen metric. The purpose of this note is t...

This paper analyses the kernel density estimation on spaces of Gaussian distributions endowed with different metrics. Explicit expressions of kernels are provided for the case of the 2-Wasserstein metric on multivariate Gaussian distributions and for the Fisher metric on multivariate centred distributions. Under the Fisher metric, the space of multivariate centred Gaussian distributions is isom...

Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ , with mean curvature τ taking all values in (−∞, 0). For n ≥ 3, define the rescaled volume of Mτ by H = |τ | Vol(M, g), where g is the induced metric. Then H ≥ nVol(M, g0) where g0 is the hyperbol...