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In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or 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 and so geometric invariants are in fact topological invariants.

See also

• Hyperbolic 3-manifold
• Margulis lemma
• Hyperbolic space