Inmathematics,hyperbolic space of dimensionn is the uniquesimply connected,n-dimensionalRiemannian manifold of constantsectional curvature equal to −1.^[1] It ishomogeneous, and satisfies the stronger property of being asymmetric space. There are many ways to construct it as an open subset of${\displaystyle {\mathbb{R}}^n}$ with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space,H², which was the first instance studied, is also called thehyperbolic plane.

It is also sometimes referred to asLobachevsky space orBolyai–Lobachevsky space after the names of the author who first published on the topic ofhyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it fromcomplex hyperbolic spaces.

Hyperbolic space serves as the prototype of aGromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of aCAT(−1) space.

The${\displaystyle n}$-dimensional hyperbolic space orhyperbolic${\displaystyle n}$-space, usually denoted${\displaystyle {\mathbb{H}}^n}$, is the unique simply connected,${\displaystyle n}$-dimensionalcomplete Riemannian manifold with a constant negative sectional curvature equal to −1.^[1] The unicity means that any two Riemannian manifolds that satisfy these properties are isometric to each other. It is a consequence of theKilling–Hopf theorem.

To prove the existence of such a space as described above one can explicitly construct it, for example as an open subset of${\displaystyle {\mathbb{R}}^n}$ with a Riemannian metric given by a simple formula. There are many such constructions or models of hyperbolic space, each suited to different aspects of its study. They are isometric to each other according to the previous paragraph, and in each case an explicit isometry can be explicitly given. Here is a list of the better-known models which are described in more detail in their namesake articles:

• Poincaré half-space model: this is the upper-half space${\displaystyle \{(x_1,\dots ,x_n)\in {\mathbb{R}}^n:x_n>0\}}$ with the metric${\displaystyle \frac{d{x}_1^2+\cdots +d{x}_n^2}{x_n^2}}$
• Poincaré disc model: this is the unit ball of${\displaystyle {\mathbb{R}}^n}$ with the metric${\displaystyle 4\frac{d{x}_1^2+\cdots +d{x}_n^2}{(1-(x_1^2+\cdots +x_n^2){)}^2}}$. The isometry to the half-space model can be realised by ahomography sending a point of theunit sphere to infinity.
• Hyperboloid model: In contrast with the previous two models this realises hyperbolic${\displaystyle n}$-space as isometrically embedded inside the${\displaystyle (n+1)}$-dimensionalMinkowski space (which is not a Riemannian but rather aLorentzian manifold). More precisely, looking at the quadratic form${\displaystyle q(x)=x_1^2+\cdots +x_n^2-x_{n+1}^2}$ on${\displaystyle {\mathbb{R}}^{n+1}}$, its restriction to the tangent spaces of the upper sheet of thehyperboloid given by${\displaystyle q(x)=-1}$ are definite positive, hence they endow it with a Riemannian metric that turns out to be of constant curvature −1. The isometry to the previous models can be realised bystereographic projection from the hyperboloid to the plane${\displaystyle \{x_{n+1}=0\}}$, taking the vertex from which to project to be${\displaystyle (0,\dots ,0,1)}$ for the ball and a point at infinity in the cone${\displaystyle q(x)=0}$ insideprojective space for the half-space.
• Beltrami–Klein model: This is another model realised on the unit ball of${\displaystyle {\mathbb{R}}^n}$; rather than being given as an explicit metric it is usually presented as obtained by using stereographic projection from the hyperboloid model in Minkowski space to its horizontal tangent plane (that is,${\displaystyle x_{n+1}=1}$) from the origin${\displaystyle (0,\dots ,0)}$.
• Symmetric space: Hyperbolic${\displaystyle n}$-space can be realised as the symmetric space of the simple Lie group${\displaystyle \mathrm{S}\mathrm{O}(n,1)}$ (the group of isometries of the quadratic form${\displaystyle q}$ with positive determinant); as a set the latter is thecoset space${\displaystyle \mathrm{S}\mathrm{O}(n,1)/\mathrm{O}(n)}$. The isometry to the hyperboloid model is immediate through the action of the connected component of${\displaystyle \mathrm{S}\mathrm{O}(n,1)}$ on the hyperboloid.

Hyperbolic space, developed independently byNikolai Lobachevsky,János Bolyai andCarl Friedrich Gauss, is a geometric space analogous toEuclidean space, but such that Euclid'sparallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):

• Given any lineL and pointP not on L, there are at least two distinct lines passing throughP that do not intersect L.

It is then a theorem that there are infinitely many such lines throughP. This axiom still does not uniquely characterize the hyperbolic plane up toisometry; there is an extra constant, the curvatureK < 0, that must be specified. However, it does uniquely characterize it up tohomothety, meaning up to bijections that only change the notion of distance by an overall constant. By choosing an appropriate length scale, one can thus assume, without loss of generality, thatK = −1.

The hyperbolic plane cannot be isometrically embedded into Euclidean 3-space byHilbert's theorem. On the other hand theNash embedding theorem implies that hyperbolic n-space can be isometrically embedded into some Euclidean space of larger dimension (5 for the hyperbolic plane by the Nash embedding theorem).

When isometrically embedded to a Euclidean space every point of a hyperbolic space is asaddle point.

The volume of balls in hyperbolic space increasesexponentially with respect to the radius of the ball rather thanpolynomially as in Euclidean space. Namely, if${\displaystyle B(r)}$ is any ball of radius${\displaystyle r}$ in${\displaystyle {\mathbb{H}}^n}$ then:$${\displaystyle \mathrm{V}\mathrm{o}\mathrm{l}(B(r))=\mathrm{V}\mathrm{o}\mathrm{l}(S^{n-1}){\int }_0^r{\sinh }^{n-1}(t)dt}$$where${\displaystyle S^{n-1}}$ is the total volume of the Euclidean${\displaystyle (n-1)}$-sphere of radius 1.

The hyperbolic space also satisfies a linearisoperimetric inequality, that is there exists a constant${\displaystyle i}$ such that any embedded disk whose boundary has length${\displaystyle r}$ has area at most${\displaystyle i\cdot r}$. This is to be contrasted with Euclidean space where the isoperimetric inequality is quadratic.

There are many more metric properties of hyperbolic space that differentiate it from Euclidean space. Some can be generalised to the setting of Gromov-hyperbolic spaces, which is a generalisation of the notion of negative curvature to general metric spaces using only the large-scale properties. A finer notion is that of a CAT(−1)-space.

Everycomplete,connected,simply connected manifold of constant negative curvature −1 isisometric to the real hyperbolic space Hⁿ. As a result, theuniversal cover of anyclosed manifoldM of constant negative curvature −1, which is to say, ahyperbolic manifold, isHⁿ. Thus, every suchM can be written asHⁿ‍/‍Γ, where Γ is atorsion-freediscrete group ofisometries onHⁿ. That is, Γ is alattice inSO⁺(n, 1).

Two-dimensional hyperbolic surfaces can also be understood according to the language ofRiemann surfaces. According to theuniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivialfundamental groupπ₁ = Γ; the groups that arise this way are known asFuchsian groups. Thequotient spaceH²‍/‍Γ of the upper half-planemodulo the fundamental group is known as theFuchsian model of the hyperbolic surface. ThePoincaré half plane is also hyperbolic, but issimply connected andnoncompact. It is theuniversal cover of the other hyperbolic surfaces.

The analogous construction for three-dimensional hyperbolic surfaces is theKleinian model.

• Dini's surface
• Hyperbolic 3-manifold
• Ideal polyhedron
• Mostow rigidity theorem
• Murakami–Yano formula
• Pseudosphere

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

• Homogeneous spaces
• Hyperbolic geometry
• Topological spaces

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