‎The aim of this paper is to establish the equivalence between the concepts‎ ‎of an $S$-metric space and a cone $S$-metric space using some topological‎ ‎approaches‎. ‎We introduce a new notion of a $TVS$-cone $S$-metric space using‎ ‎some facts about topological vector spaces‎. ‎We see that the known results on‎ ‎cone $S$-metric spaces (or $N$-cone metric spaces) can be directly obtained‎ from...

A geodesic metric space X is called hyperbolic if there exists δ ≥ 0 such that every geodesic triangle ∆ in X is δ-slim, i.e., each side of ∆ is contained in a closed δ-neighborhood of the two other sides. Let G be a group generated by a finite set A and let Γ(G,A) be the corresponding Cayley graph. The group G is said to be word hyperbolic if Γ(G,A) is a hyperbolic metric space. A subset Q of ...

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

A domain G in a Banach space is said to be δ-hyperbolic if it is a Gromov δ-hyperbolic space in the quasihyperbolic metric. Then G has the Gromov boundary ∂∗G and the norm boundary ∂G. We show that the following properties are quantitatively equivalent: (1) G is C-uniform. (2) G is δ-hyperbolic and there is a natural bijective map G ∪ ∂∗G → G ∪ ∂G, which is η-quasimöbius rel ∂∗G. (3) G is δ-hyp...

Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case i...

The main purpose of this paper is to introduce several concepts of the metric-like spaces. For instance, we define concepts such as equal-like points, cluster points and completely separate points. Furthermore, this paper is an attempt to present compatibility definitions for the distance between a point and a subset of a metric-like space and also for the distance between two subsets of a metr...

The study of holomorphic and isometric immersions of a Kähler manifold (M,g) into a Kähler manifold (N,G) started with Calabi. In his famous paper [3] he considered the case when the ambient space (N,G) is a complex space form, i.e. its holomorphic sectional curvature KN is constant. There are three types of complex space forms : flat, hyperbolic or elliptic according as the holomorphic section...

In this paper, we introduce the notion of an extended metric space ($p$-metric space) as a new generalization of the concept of $b$-metric space. Also, we present the concept of $(psi ,varphi )_{Omega}$-contractive mappings and we establish some fixed point results for this class of mappings in ordered complete $p$-metric spaces. Our results generalize several well...

Recall that the quasi-isometry group QI(X) of a metric space X is the set of equivalence classes of quasi-isometries f : X → X, where two quasiisometries f1, f2 are equivalent iff supx d(f1(x), f2(x)) <∞ (here we consider G as a metric space with a word metric). One approach to this question, which has been the most successful one, is to find an ‘optimal’ space X quasi-isometric to G and show t...

in this paper, we generalize fuzzy banach contraction theorem establishedby v. gregori and a. sapena [fuzzy sets and systems 125 (2002) 245-252]using notion of altering distance which was initiated by khan et al. [bull. austral.math. soc., 30(1984), 1-9] in metric spaces.