In mathematics, hyperbolic n-space, denoted Hⁿ, 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. Although hyperbolic space Hⁿ is diffeomorphic to Rⁿ its negative curvature metric gives it very different geometric properties. Euclid, detail from The School of Athens by Raphael. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Hyperbolic 2-space, H², is also called the hyperbolic plane. A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...

Minkowski space and hyperbolic space

Hyperbolic spaces may be regarded as models, over the real numbers, of hyperbolic geometry, that is, they satisfy the axioms of hyperbolic geometry. The principal models of hyperbolic geometry can all be related to Minkowski space. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

The Minkowski quadratic form is the form In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

In terms of Q, we may define the Minkowski bilinear form B by In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

In terms of the bilinear form, the quadratic form in turn may be defined by Q(x) = B(x,x). Rⁿ⁺¹ equipped with the Minkowski forms defines n+1 dimensional Minkowski space, R^n,1.

From the Minkowski quadratic form Q we may define a projective semialgebraic variety as the set Uⁿ defined by all x in R^n,1 such that Q(x)>0. Given two points x and y in Uⁿ, we may define a distance function by

This is a homogenous function in each coordinate, d(λx, μy) = d(x, y), for λ, μ other than zero, and so defines a function on the projective semialgebraic variety Uⁿ. This function satisfies the axioms of a metric space, and makes Uⁿ into a model of hyperbolic space, which we may consider to be a representative form of n-dimensional hyperbolic space Hⁿ. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

From this model we may derive the closely related Klein and hyperboloid models. If we choose points x in Uⁿ such that Q(x)=1, x₀ > 1, then we obtain the hyperboloid model. We may normalize to the hyperboloid model by changing the sign of x if x₀ < 0 and then dividing by . Similarly, we normalize to the Klein model by dividing x by x₀, which since Q(x)>0, cannot be zero. The Poincaré disk model can then be obtained by a slightly more elaborate mapping; see the Klein and hyperbolic model articles for mappings from these to the Poincaré disk model. The Poincaré disk model, in turn, is closely related via conformal mapping to the Poincaré half-plane model. In geometry, the Klein model, also called the projective model, the Beltrami-Klein model, the Klein-Beltrami model and the Cayley-Klein model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of... In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets. ... In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...

Another point of view on the hyperboloid model is to view it as a submanifold of (n+1)-dimensional Minkowski space, in much the same manner as the n-sphere is defined as a submanifold of (n+1)-dimensional Euclidean space. Note that the Minkowski metric tensor is not positive-definite, but rather has signature (+, −, −, …, −). This gives it rather different properties than Euclidean space. The condition x₀ > 0 selects only the top sheet of the two-sheeted hyperboloid so that Hⁿ is connected. The metric on Hⁿ is induced from the metric on R^n,1. Explicitly, the tangent space to a point x ∈ Hⁿ can be identified with the orthogonal complement of x in R^n,1. The metric on the tangent space is obtained by simply restricting the metric on R^n,1. It is important to note that the metric on Hⁿ is positive-definite even through the metric on R^n,1 is not. This means that Hⁿ is a true Riemannian manifold (as opposed to a pseudo-Riemannian manifold). This is a glossary of terms specific to differential geometry and differential topology. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

Hyperbolic manifolds

Every complete, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Uⁿ. As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is Hⁿ. Thus, every such M can be written as Hⁿ/Γ where Γ is a torsion free discrete group of isometries on Hⁿ. That is, Γ is a lattice in SO⁺(n,1). In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... The term isometric comes from the Greek for having equal measurement and is a descriptive word associated with several topics: Isometric projection, a method for the visual representation of three-dimensional objects in two dimensions, is a form of orthographic projection, or more specifically, an axonometric projection. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... See lattice for other meanings of this term, both within and without mathematics. ...

Other articles about the different models of hyperbolic space

• Hyperboloid model
• Klein model
• Poincaré disk model
• Poincaré half-plane model

In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets. ... In geometry, the Klein model, also called the projective model, the Beltrami-Klein model, the Klein-Beltrami model and the Cayley-Klein model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of... In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...

See also

• Mostow rigidity theorem
• Hyperbolic manifold
• Hyperbolic 3-manifold
• Hyperbolic geometry

In mathematics, Mostows rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group and hence unique. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannnian metric of constant sectional curvature -1. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ...

References

Ratcliffe, John G., Foundations of hyperbolic manifolds, New York, Berlin. Springer-Verlag, 1994.

Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid", American Mathematical Monthly 100:442-455. The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ...

Wolf, Joseph A. Spaces of constant curvature, 1967. See page 67.