In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a hyperbolic n-manifold is a complete Riemannian n-manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 of constant sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 -1.

Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 to the real hyperbolic space Hⁿ. As a result, the universal cover of any closed manifold M of constant negative curvature −1 is Hⁿ. Thus, every such M can be written as Hⁿ/Γ where Γ is a torsion-free discrete group of isometries onn Hⁿ. That is, Γ is a lattice in SO(R).

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean n-1-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.

For n>2 the hyperbolic structure on a finite volume hyperbolic n-manifold is unique by Mostow rigidity

Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...

 and so geometric invariants are in fact topological invariants.

See also

• Hyperbolic 3-manifold
  Hyperbolic 3-manifold
  A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
  
  
• Margulis lemma
  Margulis lemma
  In mathematics, the Margulis lemma is a result about discrete subgroups of isometries of a symmetric space , or more generally a space of non-positive curvature....
  
  
• Hyperbolic space
  Hyperbolic space
  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...
  
  
• Hyperbolization theorem
  Hyperbolization theorem
  In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.-Statement:...
  
  
• Normally hyperbolic invariant manifold
  Normally hyperbolic invariant manifold
  A normally hyperbolic invariant manifold is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described heuristically as follows: For a manifold \Lambda to be normally hyperbolic we are allowed to assume that the dynamics of \Lambda itself is...