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| [[Image:Hyperbolic orthogonal dodecahedral honeycomb.png|thumb|A perspective projection of a [[Hyperbolic small dodecahedral honeycomb|dodecahedral tessellation]] in '''[[Hyperbolic 3-manifold|H<sup>3</sup>]]'''.<BR>Four [[dodecahedron|dodecahedra]] meet at each edge, and eight meet at each vertex, like the cubes of a [[Cubic honeycomb|cubic tessellation]] in ''[[Euclidean space|E<sup>3</sup>]]'']]
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| In [[mathematics]], '''hyperbolic space''' is a type of [[non-Euclidean geometry]]. Whereas [[spherical geometry]] has a constant positive curvature, [[hyperbolic geometry]] has a negative curvature: every point in hyperbolic space is a [[saddle point]]. Parallel lines are not uniquely paired: given a line and a point not on that line, any number of lines can be drawn through the point which are coplanar with the first and do not intersect it. This contrasts with [[Euclidean geometry]], where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist, as all lines (which are great circles) cross each other. Another distinctive property is the amount of space covered by the [[n-ball]] in hyperbolic ''n''-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.
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| ==Formal definition==
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| '''Hyperbolic ''n''-space''', denoted '''H'''<sup>''n''</sup>, is the maximally symmetric, [[simply connected]], ''n''-dimensional [[Riemannian manifold]] with constant [[sectional curvature]] −1. Hyperbolic space is the principal example of a space exhibiting [[hyperbolic geometry]]. It can be thought of as the negative-curvature analogue of the ''n''-[[sphere]].
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| Although hyperbolic space '''H'''<sup>''n''</sup> is [[diffeomorphic]] to '''R'''<sup>''n''</sup> its negative-curvature metric gives it very different geometric properties.
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| Hyperbolic 2-space, '''H'''², is also called the hyperbolic plane.
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| ==Models of hyperbolic space==
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| Hyperbolic space, developed independently by Lobachevsky and Bolyai, is a geometrical space analogous to [[Euclidean space]], but such that [[parallel postulate|Euclid's parallel postulate]] is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
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| * Given any line ''L'' and point ''P'' not on ''L'', there are at least two distinct lines passing through ''P'' which do not intersect ''L''.
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| It is then a theorem that there are in fact infinitely many such lines through ''P''. Note that this axiom still does not uniquely characterize the hyperbolic plane up to [[isometry]]; there is an extra constant, the curvature ''K''<0, which must be specified. However, it does uniquely characterize it up to [[homothety]], meaning up to bijections which only change the notion of distance by an overall constant. By choosing an appropriate length-scale, one can thus assume, without loss of generality, that ''K''=-1.
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| Hyperbolic spaces are constructed in order to model such a modification of Euclidean geometry. In particular, the existence of model spaces implies that the parallel postulate is [[logically independent]] of the other axioms of Euclidean geometry.
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| There are several important models of hyperbolic space: the '''Klein model''', the '''hyperboloid model''', and the '''Poincaré model'''. These all model the same geometry in the sense that any two of them can be related by a transformation which preserves all the geometrical properties of the space. They are [[Isometry|isometric]].
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| ===The hyperboloid model===
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| {{main|Hyperboloid model}}
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| The first model realizes hyperbolic space as a hyperboloid in '''R'''<sup>n+1</sup> = {(''x''<sub>0</sub>,...,''x''<sub>n</sub>)|''x''<sub>i</sub>∈'''R''', i=0,1,...,n}. The hyperboloid is the locus '''H'''<sup>n</sup> of points whose coordinates satisfy
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| :<math>x_0^2-x_1^2-\ldots-x_n^2=1,\quad x_0>0.</math>
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| In this model a "line" (or [[geodesic]]) is the curve cut out by intersecting '''H'''<sup>n</sup> with a plane through the origin in '''R'''<sup>n+1</sup>.
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| The hyperboloid model is closely related to the geometry of [[Minkowski space]]. The [[quadratic form]]
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| :<math>Q(x) = x_0^2 - x_1^2 - x_2^2 - \cdots - x_n^2</math>
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| which defines the hyperboloid [[polarization identity|polarizes]] to give the [[bilinear form]] ''B'' defined by
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| :<math>B(x,y) = (Q(x+y)-Q(x)-Q(y))/2=x_0y_0 - x_1y_1 - \cdots - x_ny_n.</math>
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| The space '''R'''<sup>n+1</sup>, equipped with the bilinear form ''B'' is an (''n''+1)-dimensional Minkowski space '''R'''<sup>n,1</sup>.
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| From this perspective, one can associate a notion of ''distance'' to the hyperboloid model, by defining<ref>Note the similarity with the [[Elliptic geometry|chordal metric]] on a sphere, which uses trigonometric instead of hyperbolic functions.</ref> the distance between two points ''x'' and ''y'' on ''H'' to be
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| :<math>d(x, y) = \operatorname{arccosh}\, B(x,y).</math>
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| This function satisfies the axioms of a [[metric space]]. Moreover, it is preserved by the action of the [[Lorentz group]] on '''R'''<sup>n,1</sup>. Hence the Lorentz group acts as a [[transformation group]] of [[isometry|isometries]] on '''H'''<sup>n</sup>.
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| ===The Klein model===
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| {{main|Klein model}}
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| An alternative model of hyperbolic geometry is on a certain [[open set|domain]] in [[projective space]]. The Minkowski quadratic form ''Q'' defines a subset ''U''<sup>n</sup> ⊂ '''RP'''<sup>n</sup> given as the locus of points for which ''Q''(''x'') > 0 in the [[homogeneous coordinates]] ''x''. The domain ''U''<sup>n</sup> is the '''Klein model''' of hyperbolic space.
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| The lines of this model are the open line segments of the ambient projective space which lie in ''U''<sup>n</sup>. The distance between two points ''x'' and ''y'' in ''U''<sup>n</sup> is defined by
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| :<math>d(x, y) = \operatorname{arccosh}\left(\frac{B(x,y)}{\sqrt{Q(x)Q(y)}}\right).</math>
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| Note that this is well-defined on projective space, since the ratio under the inverse hyperbolic cosine is homogeneous of degree 0.
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| This model is related to the hyperboloid model as follows. Each point ''x'' ∈ ''U''<sup>n</sup> corresponds to a line ''L''<sub>x</sub> through the origin in '''R'''<sup>n+1</sup>, by the definition of projective space. This line intersects the hyperboloid '''H'''<sup>n</sup> in a unique point. Conversely, through any point on '''H'''<sup>n</sup>, there passes a unique line through the origin (which is a point in the projective space). This correspondence defines a [[bijection]] between ''U''<sup>n</sup> and '''H'''<sup>n</sup>. It is an isometry since evaluating ''d''(''x'',''y'') along ''Q''(''x'') = ''Q''(''y'') = 1 reproduces the definition of the distance given for the hyperboloid model.
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| ===The Poincaré models=== | |
| : ''Main articles: [[Poincaré disc model]], [[Poincaré half-plane model]]''
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| Another closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models. The ball model comes from a [[stereographic projection]] of the hyperboloid in '''R'''<sup>n+1</sup> onto the hyperplane {''x''<sub>0</sub> = 0}. In detail, let ''S'' be the point in '''R'''<sup>n,1</sup> with coordinates (-1,0,0,...,0): the ''South pole'' for the stereographic projection. For each point ''P'' on the hyperboloid '''H'''<sup>n</sup>, let ''P''<sup>*</sup> be the unique point of intersection of the line ''SP'' with the plane {''x''<sub>0</sub> = 0}. This establishes a bijective mapping of '''H'''<sup>n</sup> into the unit ball
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| :<math> B^n = \{(x_1,\ldots,x_n) | x_1^2+\ldots+x_n^2 < 1\}</math>
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| in the plane {''x''<sub>0</sub> = 0}.
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| The geodesics in this model are [[semicircle]]s which are perpendicular to the boundary sphere of ''B''<sup>n</sup>. Isometries of the ball are generated by [[inversive geometry|spherical inversion]] in hyperspheres perpendicular to the boundary. | |
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| The half-space model results from applying an [[inversion in a point]] of the boundary of ''B''<sup>n</sup>. This sends circles to circles and lines, and is moreover a [[conformal transformation]]. Consequently the geodesics of the half-space model are lines and circles perpendicular to the boundary hyperplane.
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| ==Hyperbolic manifolds== | |
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| Every [[complete space|complete]], [[connected space|connected]], [[simply-connected]] manifold of constant negative curvature −1 is [[Isometry|isometric]] to the real hyperbolic space '''H'''<sup>''n''</sup>. As a result, the [[universal cover]] of any [[closed manifold]] ''M'' of constant negative curvature −1, which is to say, a [[hyperbolic manifold]], is '''H'''<sup>''n''</sup>. Thus, every such ''M'' can be written as '''H'''<sup>''n''</sup>/Γ where Γ is a [[torsion (algebra)|torsion-free]] [[discrete group]] of [[isometry|isometries]] on '''H'''<sup>''n''</sup>. That is, Γ is a [[lattice (discrete subgroup)|lattice]] in SO<sup>+</sup>(''n'',1).
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| === Riemann surfaces ===
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| Two-dimensional hyperbolic surfaces can also be understood according to the language of [[Riemann surface]]s. According to the [[uniformization theorem]], every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial [[fundamental group]] π<sub>1</sub>=Γ; the groups that arise this way are known as [[Fuchsian group]]s. The [[quotient space]] '''H'''²/Γ of the upper half-plane [[Ideal (ring theory)|modulo]] the fundamental group is known as the [[Fuchsian model]] of the hyperbolic surface. The [[Poincaré half plane]] is also hyperbolic, but is [[simply connected]] and [[noncompact]]. It is the [[universal cover]] of the other hyperbolic surfaces.
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| The analogous construction for three-dimensional hyperbolic surfaces is the [[Kleinian model]].
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| ==See also==
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| * [[Hyperbolic geometry]]
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| * [[Mostow rigidity theorem]]
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| * [[Hyperbolic manifold]]
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| * [[Hyperbolic 3-manifold]]
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| * [[Murakami–Yano formula]]
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| * [[Pseudosphere]]
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| * [[Dini's surface]]
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| ==References==
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| {{reflist}}
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| *{{aut|A'Campo, Norbert and Papadopoulos, Athanase}}, (2012) ''Notes on hyperbolic geometry'', in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.
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| * Ratcliffe, John G., ''Foundations of hyperbolic manifolds'', New York, Berlin. Springer-Verlag, 1994.
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| * Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid", [[American Mathematical Monthly]] 100:442-455.
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| * Wolf, Joseph A. ''Spaces of constant curvature'', 1967. See page 67.
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| * [http://arxiv.org/abs/0903.3287 Hyperbolic Voronoi diagrams made easy, Frank Nielsen]
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| [[Category:Homogeneous spaces]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Topological spaces]]
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