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searching for Poincaré metric 9 found (39 total)

Schwarz–Ahlfors–Pick theorem (214 words) [view diff] exact match in snippet view article find links to article

increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised

Complex geodesic (345 words) [view diff] exact match in snippet view article find links to article

2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by ρ ( a , b ) = tanh − 1 ⁡ | a − b | | 1 − a ¯ b |

L² cohomology (541 words) [view diff] exact match in snippet view article find links to article

"Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric". Annals of Mathematics. 109 (3): 415–476. doi:10.2307/1971221. JSTOR 1971221

Ahlfors finiteness theorem (326 words) [view diff] exact match in snippet view article find links to article

with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then

Steven Zucker (345 words) [view diff] exact match in snippet view article find links to article

"Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric". Annals of Mathematics. 109 (3): 415–476. doi:10.2307/1971221. JSTOR 1971221

De Sitter space (2,198 words) [view diff] exact match in snippet view article find links to article

Encyclopedia of Mathematics, EMS Press Nomizu, Katsumi (1982), "The Lorentz–Poincaré metric on the upper half-space and its extension", Hokkaido Mathematical Journal

Tessellation (6,042 words) [view diff] exact match in snippet view article find links to article

course this is not true for the Euclidean metric, but holds for the Poincaré metric Leys, Jos (2015). "Hyperbolic Escher". Retrieved 27 May 2015. Escher

Busemann function (12,928 words) [view diff] exact match in snippet view article find links to article

D {\displaystyle D} be the unit disk in the complex plane with the Poincaré metric d s 2 = 4 | d z | 2 ( 1 − | z | 2 ) 2 . {\displaystyle ds^{2}={4\,|dz|^{2}

Schwarzian derivative (6,945 words) [view diff] exact match in snippet view article find links to article

\log[(1+|\mathbf {x} |^{2})^{-1}]} , or else a conformal factor for a hyperbolic Poincaré metric on the ball or half-space log ⁡ | ( 1 − | x | 2 ) − 1 | {\displaystyle