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Space where every point locally resembles a hyperbolic space
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in
Hyperbolic_manifold
Manifold of dimension 3 equipped with a hyperbolic metric
topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric
Hyperbolic_3-manifold
Spacetime manifold
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
Globally_hyperbolic_spacetime
Mathematical space
geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful. The fundamental groups of 3-manifolds strongly reflect
3-manifold
Smooth manifold with an inner product on each tangent space
-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian
Riemannian_manifold
Non-Euclidean geometry
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature
Hyperbolic_space
instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
Theorem in 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
Mostow_rigidity_theorem
Topological complexity in mathematics
proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume. The simplicial volume is equal to twice
Simplicial_volume
Three-holed sphere
; Harvey, William J.; Recillas-Pishmish, Sevín (eds.). Complex Manifolds and Hyperbolic Geometry. Contemporary Mathematics. Vol. 311. Providence, RI: American
Pair_of_pants_(mathematics)
Smallest closed orientable hyperbolic 3-manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1)
Weeks_manifold
Formalization of the idea of an attractor or repellor in dynamical systems
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
Stable_manifold
Normalized hyperbolic volume of the complement of a hyperbolic knot
constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously
Hyperbolic_volume
Theorem in geometry
or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. One form of
Hyperbolization_theorem
Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one
Seifert–Weber_space
A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described
Normally hyperbolic invariant manifold
Normally_hyperbolic_invariant_manifold
the complex hyperbolic space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex
Complex_hyperbolic_space
Fixed point that does not have any center manifolds
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
Hyperbolic_equilibrium_point
hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn
Hyperbolic_Dehn_surgery
In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume, discovered by Hugo Gieseking. It is non-orientable and has the
Gieseking_manifold
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
Margulis_lemma
Result in dynamical systems theory
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
Stable_manifold_theorem
Type of non-Euclidean geometry
model Constructions in hyperbolic geometry Hjelmslev transformation Hyperbolic 3-manifold Hyperbolic manifold Hyperbolic set Hyperbolic tree Kleinian group
Hyperbolic_geometry
dynamical systems theory, a subset Λ of a smooth manifold M is said to be hyperbolic or to have a hyperbolic structure with respect to a smooth map f if its
Hyperbolic_set
Concept in hyperbolic geometry
In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by William Thurston (1986). Given a
Earthquake_map
Differentiable manifold with nondegenerate metric tensor
mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere
Pseudo-Riemannian_manifold
Pseudometric of complex manifolds
complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined
Kobayashi_metric
Topological Object
put on the set H of hyperbolic 3-manifolds of finite volume. Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental
Geometric_topology_(object)
eleventh problem out of his twenty-four questions, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their
Ending_lamination_theorem
Topological space that locally resembles Euclidean space
gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant
Manifold
mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space H 3 / Γ {\displaystyle \mathbb {H} ^{3}/\Gamma
Kleinian_model
In mathematics, a Riemann surface
S2CID 14305255. Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math. Vol. 219. New York: Springer. ISBN 0-387-98386-4
Bolza_surface
Mathemical concept
In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ( 5 , 1 ) {\displaystyle (5,1)} surgery on the figure-8
Meyerhoff_manifold
Shape in hyperbolic geometry
an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique
Ideal_polyhedron
describe basic background material on hyperbolic geometry. Chapter 4 covers Dehn surgery on hyperbolic manifolds Chapter 5 covers results related to Mostow's
The geometry and topology of three-manifolds
The_geometry_and_topology_of_three-manifolds
of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically
Geometric_finiteness
Hyperbolic 3-manifold proposed as a model for the shape of the universe
known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space
Picard_horn
Type of mathematical link
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
Hyperbolic_link
Foundational examples are hyperbolic manifolds and affine manifolds. Let X {\displaystyle X} be a connected differentiable manifold and G {\displaystyle G}
(G,_X)-manifold
Fundamental result in geometry
Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149, Springer, p. 99, ISBN 9780387331973, That the area of a hyperbolic triangle is
Sum_of_angles_of_a_triangle
groups of complete noncompact hyperbolic manifolds of finite volume. Further generalizations such as acylindrical hyperbolicity are also explored by current
Relatively_hyperbolic_group
No complete regular surface of constant negative gaussian curvature immerses in R3
:H\rightarrow S'} will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold S ′ {\displaystyle S'} , which carries the inner
Hilbert's theorem (differential geometry)
Hilbert's_theorem_(differential_geometry)
Discrete group of Möbius transformations
{\displaystyle \pi _{1}} of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold. Many authors[who?] use the terms
Kleinian_group
Compact Riemann surface of genus 3
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism
Klein_quartic
Manifold with Riemannian, complex and symplectic structure
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a
Kähler_manifold
Russian mathematician (born 1966)
Union portal 50033 Perelman Ancient solution Homology sphere Hyperbolic manifold "Manifold Destiny" Spherical space form conjecture Thurston elliptization
Grigori_Perelman
Three dimensional analogue of uniformization conjecture
stabilizer is O(2,R). Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the
Geometrization_conjecture
Three linked but pairwise separated rings
in 1991 by the Geometry Center. Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition)
Borromean_rings
One-dimensional complex manifold
function-theoretic classification but it is hyperbolic in the geometric classification. Dessin d'enfant Kähler manifold Lorentz surface Mapping class group Serre
Riemann_surface
Two interlinked loops with five structural crossings
respectively one of the minimum-volume hyperbolic manifolds with one cusp and the minimum-volume hyperbolic manifold with no cusps. The Whitehead link is
Whitehead_link
Mathematical software
mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version
SnapPea
Type of geometry in mathematics
Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental
Ricci-flat_manifold
Branch of mathematics
geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of vector calculus, linear algebra and multilinear
Differential_geometry
Mathematical concept
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_group
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1)
Hurwitz_surface
American mathematician (1946–2012)
most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem. To complete
William_Thurston
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame
Tameness_theorem
Mathematical space
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four,
4-manifold
Diffeomorphism that has a hyperbolic structure on the tangent bundle
fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked
Anosov_diffeomorphism
Unique knot with a crossing number of four
volume among non-compact hyperbolic 3-manifolds. The figure-eight knot and the (−2,3,7) pretzel knot are the only two hyperbolic knots known to have more
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Type of manifold
information matrix, it is a statistical manifold with a geometry modeled on hyperbolic space. A way of picturing the manifold is done by inferring the parametric
Statistical_manifold
American mathematician
Scottish-American mathematician working primarily with arithmetic hyperbolic 3-manifolds. He is the Edgar Odell Lovett Chair of mathematics at Rice University
Alan_Reid_(mathematician)
Causal relationships between points in a manifold
structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according
Causal_structure
In Riemann surface theory and hyperbolic geometry, the MacBeath surface, also called MacBeath curve or the Fricke–MacBeath surface curve, is the genus-7
MacBeath_surface
24 mathematical problems stated in 1982
Thurston in his influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical
Thurston's_24_questions
Type of curve in geometry
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic: one whose parametrization is not obtained by going repeatedly
Prime_geodesic
Riemannian flat manifold. Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H3, hence aspherical. As is any n-manifold whose universal
Aspherical_space
Partial differential equation
E3, three-dimensional hyperbolic space H3, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous
Ricci_flow
Locally defined function in general relativity
Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a globally hyperbolic manifold, and in the definition of Hadamard
Synge's_world_function
American mathematician
considered the case of a closed hyperbolic 3-manifold M that fibers over the circle with the fiber being a closed hyperbolic surface S. In this case the universal
James_W._Cannon
Way to divide polygon into smaller parts
architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule
Finite_subdivision_rule
Algebraic surface
curve can be obtained as a Riemann surface by associating sides of a hyperbolic icosagon (see fundamental polygon). The identification pattern is given
Bring's_curve
Value determined from a polyhedron
reassembled into each other. Every hyperbolic manifold with finite volume can be cut along geodesic surfaces into a hyperbolic polyhedron (a fundamental domain
Dehn_invariant
Manifold upon which it is possible to perform calculus
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow
Differentiable_manifold
Triangle in hyperbolic geometry
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
Hyperbolic_triangle
Concept in mathematics
genus is hyperbolic (it is in fact quasi-isometric to the hyperbolic plane). The hyperbolic plane (and more generally any Hadamard manifolds of sectional
Hyperbolic_metric_space
not tame. Closed manifold – Topological concept in mathematics Tameness theorem Gabai, David (2009), "Hyperbolic geometry and 3-manifold topology", in Mrowka
Tame_manifold
generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that
Non-positive_curvature
Indian mathematician and monk of the Ramakrishna Order (born 1968)
He has widely published and presented his research in the area of hyperbolic manifolds and ending lamination spaces. His most notable work is the proof
Mahan_Mj
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
(H_{i})} is the hyperbolic volume of the hyperbolic manifold H i {\displaystyle H_{i}} . As a special case, if K {\displaystyle K} is a hyperbolic knot, then
Volume_conjecture
Danish mathematician
Jørgensen's inequality, and for his discovery of a hyperbolic structure on certain fibered 3-manifolds which were one of the inspirations for William Thurston's
Troels_Jørgensen
Length of a line segment
1090/S0273-0979-1982-14958-8, MR 0634431 Ratcliffe, John G. (2019), Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149 (3rd ed.), Springer, p. 32
Euclidean_distance
American mathematician
surfaces, quadratic differentials, Teichmüller spaces, hyperbolic geometry of surfaces and 3-manifolds, Fuchsian groups, Kleinian groups, complex dynamics
Albert_Marden
many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale
Morse–Smale_system
Branch of topology
studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the theory of 3-manifolds and 4-manifolds, knot
Low-dimensional_topology
Japanese mathematician
pseudo-metric to any complex manifold, in a holomorphically invariant way. This sets up the important notion of Kobayashi hyperbolicity, which is defined by the
Shoshichi_Kobayashi
Type of Riemannian manifold with constant Jacobi operator spectrum
the Cayley hyperbolic plane C a y H 2 {\displaystyle \mathbb {C} ayH^{2}} . Clifford structures are fundamental in studying Osserman manifolds. An algebraic
Osserman_manifold
Mathematical concept
Invariant manifold Stable manifold Lagrangian coherent structure Normally hyperbolic invariant manifold Roberts, A.J. (1993). "The invariant manifold of beam
Center_manifold
Riemannian manifold which satisfies vacuum Einstein equations
the relationship between spheres and hyperbolic spaces. One necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe
Einstein_manifold
Parametrizes complex structures on a surface
Since Teichmüller space is a complex manifold it carries a Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its Kobayashi metric coincides
Teichmüller_space
Quotient of a weakly contractible space by a free action
_{1}(S).} A closed (that is, compact and without boundary) connected hyperbolic manifold M is a classifying space for its fundamental group π 1 ( M ) {\displaystyle
Classifying_space
reformulation of Mostow rigidity, weakening the hypothesis from hyperbolic manifolds to aspherical manifolds, and similarly weakening the conclusion from an isometry
Borel_conjecture
Topological manifold that is invariant under the action of dynamical system
manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold. Typically, although by no means always, invariant manifolds are constructed
Invariant_manifold
Russian-French mathematician
group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the notion of almost flat manifolds.[G78] The famous quarter-pinched
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Property of measure-preserving dynamical systems
curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure); the horocycle flow on a hyperbolic manifold of finite volume
Ergodicity
2D surface which extends indefinitely
the real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature. Abstractly, one
Plane_(mathematics)
Branch of mathematics that studies dynamical systems
development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in
Ergodic_theory
American mathematician
s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas for special
Don_Zagier
geometrization conjecture Hyperbolic 3-manifolds Spherical 3-manifolds Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups
List of geometric topology topics
List_of_geometric_topology_topics
Property of a mathematical space
topological manifold can be calculated. A connected topological manifold is locally homeomorphic to Euclidean n-space, in which the number n is the manifold's dimension
Dimension
travel, tourism, insurance
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
Boy/Male
Hindu, Indian
Manifoldness; Variety
Girl/Female
Hindu, Indian, Marathi, Sanskrit, Tamil
Manifold; Variegated
Surname or Lastname
English
English : unexplained. It may be a variant of Minnifield, which is likewise unexplained.
Boy/Male
Indian, Sanskrit
Manifold; Multiplied
Boy/Male
Indian, Sanskrit
Plenty; Much; Strong; Manifold
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
HYPERBOLIC MANIFOLD
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