In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in...

topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric...

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...

-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all Riemannian manifolds. Riemannian...

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...

geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful. The fundamental groups of 3-manifolds strongly reflect...

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...

with e Hyperbolic manifold, a complete Riemannian n-manifold of constant sectional curvature −1 Hyperbolic motion, an isometry in a hyperbolic space Hyperbolic...

instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an...

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere...

model Constructions in hyperbolic geometry Hjelmslev transformation Hyperbolic 3-manifold Hyperbolic manifold Hyperbolic set Hyperbolic tree Kleinian group...

A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described...

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...

{\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...

Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one...

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...

essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group...

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...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. One form of...

Foundational examples are hyperbolic manifolds and affine manifolds. Let X {\displaystyle X} be a connected differential manifold and G {\displaystyle G}...

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...

precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...

hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn...

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...

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...

an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique...

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...

portal Ancient solution Asteroid 50033 Perelman Homology sphere Hyperbolic manifold "Manifold Destiny" (On The New Yorker article) Spherical space form conjecture...

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...