Hyperbolic Geometry

Anderson, James A.

doi:10.1007/978-1-4471-3987-4

Cited by 142 publications

(228 citation statements)

References 8 publications

(14 reference statements)

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“…Theorem 2.9 implies that (2) is equivalent to (5) and that (3) is equivalent to (6). It is clear that (2) implies (3), with a 0 = a and b 0 = b.…”

Section: Results In Metric Spacesmentioning

confidence: 96%

“…In particular, characterization (5) gives that it is sufficient to check the Rips condition just for bigons.…”

Section: Results In Riemann Surfacesmentioning

confidence: 99%

“…Every hexagon in the unit disk is 4 log(1 + √ 2 )-thin, since the unit disk is log(1 + √ 2 )-thin (see, e.g. [5], p.130). Let us denote by w a point in ∂H n0 \ γ + n0+1 with d Ω (z, w) ≤ 4 log(1 + √ 2 ).…”

Section: Results In Riemann Surfacesmentioning

confidence: 99%

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Gromov hyperbolicity of Denjoy Domains

et al. 2006

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Abstract. In this paper we characterize the Gromov hyperbolicity of the double of a metric space. This result allows to give a characterization of the hyperbolic Denjoy domains, in terms of the distance to R of the points in some geodesics. In the particular case of trains (a kind of Riemann surfaces which includes the flute surfaces), we obtain more explicit criteria which depend just on the lengths of what we have called fundamental geodesics.

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“…Theorem 2.9 implies that (2) is equivalent to (5) and that (3) is equivalent to (6). It is clear that (2) implies (3), with a 0 = a and b 0 = b.…”

Section: Results In Metric Spacesmentioning

confidence: 96%

“…In particular, characterization (5) gives that it is sufficient to check the Rips condition just for bigons.…”

Section: Results In Riemann Surfacesmentioning

confidence: 99%

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Gromov hyperbolicity of Denjoy Domains

et al. 2006

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“…We say that a complex valued random field {X t , t ∈ T } is a process with stationary increments in a wide sense starting at o if X o = 0, EX t = 0 for all t ∈ T and ∀t 1 …”

Section: Fields With Stationary Incrementsmentioning

confidence: 99%

Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres

Cohen¹,

Lifshits²

2012

ESAIM: PS

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Abstract.We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.Mathematics Subject Classification. 60G15, 60G10, 51M10.

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“…In this half plane geometry, the lines and the function of distance are different, therefore, it seems interesting to study the Poincaré analogues of the topics that include the concept of distance in the Euclidean geometry. A few of such topics have been studied by some authors [1,[3][4][5][6][7][8]. In this work, it is shown that the coordinates of the division point can be determined by the formula in the Poincaré upper half plane.…”

Section: Introductionmentioning

confidence: 99%

Division Point in the Poincaré Upper Half Plane

Sönmez¹

2009

Demonstratio Mathematica

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Abstract. In this work, it is shown that the coordinates of the division point can be determined by the formula in the Poincaré upper half plane. IntroductionThe Poincaré upper half plane geometry has been introduced by Henri The geometry of the half plane H is a non-Euclidean, since it fails to satisfy the parallel postulate but satisfies all the remaining twelve axioms of 1991 Mathematics Subject Classification: 51K05, 51K99.

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Hyperbolic Geometry

Cited by 142 publications

References 8 publications

Gromov hyperbolicity of Denjoy Domains

Gromov hyperbolicity of Denjoy Domains

Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres

Division Point in the Poincaré Upper Half Plane

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