Let S be the unit sphere in the Euclidean space R, and let e be the standard metric on S induced from R. Suppose that (u, ρ) are the spherical coordinates in R, where u ∈ S, ρ ∈ [0,∞). By choosing the smooth function φ(ρ) := sinh ρ on [0,∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S, 0 ≤ ρ < ∞} as follows h = dρ + φ(ρ)e. This gives the space form R(−1) which is the hyperboli...

For every metric space X, we deene a continuous poset BX such that X is home-omorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo ii X is complete, and !-continuous ii X is separable. The computational model BX is used to give domain-theoretic proofs of Banach's xed point theorem and of two classical results of Hutchinson: on a complete metric s...

It is shown that optical geometry of the Reissner-Nordström exterior metric can be embedded in a hyperbolic space all the way down to its outer horizon. The adopted embedding procedure removes a breakdown of flat-space embeddings which occurs outside the horizon, at and below the Buchdahl-Bondi limit (R/M = 9/4 in the Schwarzschild case). In particular, the horizon can be captured in the optica...

The Farrell-Jones warping deformation is a powerful geometric construction that has been crucial in the proofs of many important contributions to the theory of manifolds of negative curvature. In this paper we study this construction in depth, in a more general setting, and obtain explicit quantitative results. The results in this paper are key ingredients in the problem of smoothing Charney-Da...

We use cross ratios to describe second real continuous bounded cohomology for locally compact σ-compact topological groups. We also investigate the second continuous bounded cohomology group of a closed subgroup of the isometry group Iso(X) of a proper hyperbolic geodesic metric space X and derive some rigidity results for Iso(X)-valued cocycles.

We focus on the study of time-varying paths in the two-dimensional hyperbolic space, and our aim is to define a reparameterization invariant distance on the space of such paths. We adapt the geodesical distance on the space of parameterized plane curves given by Bauer et al. in [1] to the space Imm([0, 1],H) of parameterized curves in the hyperbolic plane. We present a definition which enables ...

For any hyperbolic complex X and a ∈ X we construct a visual metric ď = ďa on ∂X that makes the Isom(X)-action on ∂X bi-Lipschitz, Möbius, symmetric and conformal. We define a stereographic projection of ďa and show that it is a metric conformally equivalent to ďa. We also introduce a notion of hyperbolic dimension for hyperbolic spaces with group actions. Problems related to hyperbolic dimensi...

We consider a nearly hyperbolic Sasakian manifold endowed with a quarter symmetric non-metric connection and study CRsubmanifolds of nearly hyperbolic Sasakian manifold endowed with a quarter symmetric nonmetric connection. We also obtain parallel distributions and discuss inerrability conditions of distributions on CR-submanifolds of nearly hyperbolic Sasakian manifold with quarter symmetric n...

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are strongly hyperbolic. On the way, we determine the best ...

We show that if a complete, doubling metric space is annulus linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, hyperbolic groups whose boundaries have no local cut points have conformal dimension greater than one; this answers a question of Bonk and Kleiner.