point of a hyperbolic space is a saddle point. Volume growth and isoperimetric inequality The volume of balls in hyperbolic space increases exponentially with respect to the radius of the ball rather than polynomially as in Euclidean space. Namely, if is any ball of radius in then: where is the total volume of the Euclidean -sphere of radius 1. The hyperbolic space also satisfies a linear isoperimetric inequality, that is there exists a constant such that any embedded disk whose boundary has length has area at most . This is to be contrasted with Euclidean space where the isoperimetric inequality is quadratic. Other metric properties There are many more metric properties of hyperbolic space which 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. Hyperbolic manifolds Every complete, connected, simply connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. 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). Riemann surfaces Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π1=Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H²/Γ of the upper half-plane 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. The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model. See also Dini's surface Hyperbolic 3-manifold Ideal polyhedron Mostow rigidity theorem Murakami–Yano formula Pseudosphere 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. Wolf, Joseph A. Spaces of constant curvature, 1967. See page 67. Homogeneous spaces Hyperbolic geometry Topological spaces