We introduce the quasi-hyperbolicity constant of a metric space, rough isometry invariant that measures how space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in closed interval $[1,2]$. The an hyperbolic is equal to one. For CAT$(0)$-space, it bounded above by $\sqrt{2}$. Banach at least two dimensional below $\sqrt{2}$, and non-trivial $L_p$-space exactly $\m...

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δneighborhood of the union of the other two sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) ...

A bstract The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric charge two SU(2) monopoles in Euclidean space. Here a hyperbolic analogue presented, by calculating boundary metric on calculation performed using twistor description and result presented terms standard elliptic integrals.

Let M be a compact 4-manifold with boundary ∂M , and consider the moduli space EAH of asymptotically hyperbolic Einstein (AHE) metrics on M . Any such metric g induces a conformal class [γ] of metrics on ∂M , called the conformal infinity of g. In this paper, we prove that the space of boundary values BAH of metrics in EAH is closed under natural conditions, i.e. if [γi] is a sequence of confor...

We define a group to be translation discrete if it carries a metric in which the translation numbers of the non-torsion elements are bounded away from zero. We define the notion of quasiconvex space which generalizes the notion of both CAT(0) and Gromov–hyperbolic spaces. We show that a cocompact group of isometries acting properly discontinuously cocompactly on a proper quasiconvex metric spac...

We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp ...

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε0-relative ǫ-thick p...

– (p, q) (p > 2, q = 2). These infinite tiling series of cubes are the special cases of the classical Lambert-cube tilings. The dihedral angles of the Lambert-cube are πp (p > 2) at the 3 skew edges and π2 at the other edges. Their metric realization in the hyperbolic space H 3 is well known. A simple proof was described by E. Molnár in [10]. The volume of this Lambert-cube type was determined ...

in the present paper, we give a new approach to caristi's fixed pointtheorem on non-archimedean fuzzy metric spaces. for this we define anordinary metric $d$ using the non-archimedean fuzzy metric $m$ on a nonemptyset $x$ and we establish some relationship between $(x,d)$ and $(x,m,ast )$%. hence, we prove our result by considering the original caristi's fixedpoint theorem.