In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in...
6 KB (680 words) - 02:46, 5 July 2023
global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It is called hyperbolic in analogy...
9 KB (1,349 words) - 12:20, 20 October 2023
topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric...
16 KB (2,217 words) - 15:58, 22 June 2024
-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all Riemannian manifolds. Riemannian...
60 KB (8,581 words) - 14:46, 19 July 2024
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...
11 KB (1,538 words) - 00:57, 19 April 2024
geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful. The fundamental groups of 3-manifolds strongly reflect...
45 KB (5,836 words) - 09:37, 24 April 2024
unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics...
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with e Hyperbolic manifold, a complete Riemannian n-manifold of constant sectional curvature −1 Hyperbolic motion, an isometry in a hyperbolic space Hyperbolic...
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instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an...
12 KB (1,714 words) - 07:38, 29 March 2024
mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere...
9 KB (1,164 words) - 03:07, 18 June 2024
model Constructions in hyperbolic geometry Hjelmslev transformation Hyperbolic 3-manifold Hyperbolic manifold Hyperbolic set Hyperbolic tree Kleinian group...
56 KB (6,993 words) - 15:18, 23 June 2024
A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described...
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systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the...
5 KB (645 words) - 04:28, 29 February 2024
Seifert–Weber space (redirect from Hyperbolic dodecahedral space)
Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one...
4 KB (462 words) - 05:08, 29 March 2024
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...
21 KB (2,753 words) - 18:41, 26 January 2024
the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits...
54 KB (8,837 words) - 05:09, 15 July 2024
Mostow rigidity theorem (category Hyperbolic geometry)
essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group...
8 KB (1,068 words) - 06:35, 2 April 2024
Kleinian group (category 3-manifolds)
{\displaystyle \pi _{1}} of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold. Many authors use the terms Kleinian...
18 KB (2,282 words) - 20:19, 14 June 2024
many more hyperbolic 3-manifolds. Borromean rings are hyperbolic. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result...
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In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...
33 KB (4,736 words) - 09:41, 5 June 2024
Hyperbolization theorem (category Hyperbolic geometry)
or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. One form of...
9 KB (1,048 words) - 21:31, 10 July 2023
equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly...
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hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn...
6 KB (845 words) - 08:04, 23 May 2024
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three...
13 KB (1,759 words) - 18:14, 1 April 2024
Foundational examples are hyperbolic manifolds and affine manifolds. Let X {\displaystyle X} be a connected differential manifold and G {\displaystyle G}...
8 KB (1,378 words) - 20:19, 21 September 2022
mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex...
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Ideal polyhedron (redirect from Ideal hyperbolic polyhedron)
an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique...
27 KB (3,216 words) - 15:08, 4 January 2024
Margulis lemma (category Hyperbolic geometry)
subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-space). Roughly, it states that within a fixed radius, usually...
8 KB (1,185 words) - 18:50, 26 January 2024
portal Ancient solution Asteroid 50033 Perelman Homology sphere Hyperbolic manifold "Manifold Destiny" (On The New Yorker article) Spherical space form conjecture...
66 KB (6,459 words) - 16:32, 12 July 2024
In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle...
3 KB (508 words) - 20:01, 29 October 2023