Hyperbolic 3-manifold

Summary

A hyperbolic $3$-manifold is a 3-manifold with a complete Riemannian metric of constant curvature $-1$. Equivalently, it is a quotient of hyperbolic $3$-space $\mathbb{H}^3$ by a discrete torsion-free group of isometries, a Kleinian group in the orientable case. By the geometrization theorem, hyperbolic manifolds form the largest and richest class of $3$-manifolds. By Mostow rigidity, a hyperbolic metric of finite volume is determined by the fundamental group, so geometric quantities such as volume are topological invariants.