Statements (26)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
gptkb:unit_disk
upper half-plane |
| gptkbp:category |
gptkb:Riemannian_geometry
gptkb:hyperbolic_geometry |
| gptkbp:conformal |
yes
|
| gptkbp:curvature |
constant negative curvature
|
| gptkbp:curvatureValue |
-1
|
| gptkbp:distance |
arccosh(1 + 2|z-w|^2/((1-|z|^2)(1-|w|^2))) for unit disk
arccosh(1 + |z-w|^2/(2 Im(z) Im(w))) for upper half-plane |
| gptkbp:field |
complex analysis
differential geometry |
| gptkbp:hasInvariant |
gptkb:Möbius_transformations
|
| https://www.w3.org/2000/01/rdf-schema#label |
Poincaré metric
|
| gptkbp:metricTensor |
ds^2 = 4|dz|^2/(1-|z|^2)^2 on unit disk
ds^2 = |dz|^2/(Im(z))^2 on upper half-plane |
| gptkbp:namedAfter |
gptkb:Henri_Poincaré
|
| gptkbp:provides |
model of hyperbolic plane
|
| gptkbp:relatedTo |
gptkb:Fuchsian_groups
automorphic forms |
| gptkbp:type |
gptkb:Riemannian_manifold
|
| gptkbp:usedIn |
gptkb:hyperbolic_geometry
gptkb:Teichmüller_theory complex manifolds |
| gptkbp:bfsParent |
gptkb:Henri_Poincaré
|
| gptkbp:bfsLayer |
5
|