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... > topology > algebraic topology > manifold > differentiable manifold > complex manifold > Riemann surface > Poincaré metric
... > geometry > topological space > manifold > differentiable manifold > complex manifold > Riemann surface > Poincaré metric
... > geometry > differential geometry > differentiable manifold > complex manifold > Riemann surface > Poincaré metric
... > topology > differential topology > differentiable manifold > complex manifold > Riemann surface > Poincaré metric
geometry > complex geometry > complex manifold > Riemann surface > Poincaré metric
geometry > non-Euclidean geometry > hyperbolic geometry > Poincaré metric
geometry > differential geometry > tensor > metric tensor > Poincaré metric
algebra > multilinear algebra > tensor > metric tensor > Poincaré metric

Preferred term

Poincaré metric  

Definition

  • In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Poincar%C3%A9_metric)

Broader concept

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http://data.loterre.fr/ark:/67375/PSR-HMHLXJHR-5

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