---
tag: GHVW1Z35BS
kind: concept
name: Hyperbolic 3-manifold
msc: [57K32]
specializes: [R20TB2KEQQ, 7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_3-manifold
same_as:
  wikidata: Q5957770
  wikipedia: Hyperbolic 3-manifold
---

## Summary

A hyperbolic $3$-manifold is a [[3-manifold|R20TB2KEQQ]] 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.

## Explanations

### L1 human

Hyperbolic space is a space where triangles have angle sum less than $\pi$ and where volume grows exponentially with distance.
A hyperbolic $3$-manifold is a space that looks like hyperbolic space near every point.
One can build one by gluing the faces of a polyhedron in hyperbolic space in pairs, so that the angles around each edge add up to a full turn.
The complement of the figure-eight knot in $S^3$ is obtained by gluing two ideal tetrahedra, and it is hyperbolic.