{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "3EVAF9NG5Z"
    },
    {
      "field": "about",
      "target": "BB28P2ED5H"
    },
    {
      "field": "about",
      "target": "DE1Z7R9T05"
    },
    {
      "field": "uses",
      "target": "3EVAF9NG5Z"
    },
    {
      "field": "uses",
      "target": "BB28P2ED5H"
    },
    {
      "field": "uses",
      "target": "DE1Z7R9T05"
    },
    {
      "field": "uses",
      "target": "DYFQ2EQ7J6"
    },
    {
      "field": "uses",
      "target": "ZP4AWD45CE"
    },
    {
      "field": "instance_of",
      "target": "3HDRK3QYSD"
    },
    {
      "field": "instance_of",
      "target": "FBVEG8HVTP"
    },
    {
      "field": "defines",
      "target": "BNYW1SNP00"
    }
  ],
  "body": "\n## Summary\n\nA hyperbolic $3$-manifold is a [[3-manifold|R20TB2KEQQ]] with a complete Riemannian metric of constant curvature $-1$.\nEquivalently, 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.\nBy the geometrization theorem, hyperbolic manifolds form the largest and richest class of $3$-manifolds.\nBy Mostow rigidity, a hyperbolic metric of finite volume is determined by the fundamental group, so geometric quantities such as volume are topological invariants.\n\n## Explanations\n\n### L1 human\n\nHyperbolic space is a space where triangles have angle sum less than $\\pi$ and where volume grows exponentially with distance.\nA hyperbolic $3$-manifold is a space that looks like hyperbolic space near every point.\nOne 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.\nThe complement of the figure-eight knot in $S^3$ is obtained by gluing two ideal tetrahedra, and it is hyperbolic.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "57K32"
    ],
    "name": "Hyperbolic 3-manifold",
    "same_as": {
      "wikidata": "Q5957770",
      "wikipedia": "Hyperbolic 3-manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyperbolic_3-manifold"
      }
    ],
    "specializes": [
      "R20TB2KEQQ",
      "7CBT4SM83Y"
    ],
    "tag": "GHVW1Z35BS"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/GHVW1Z35BS/",
  "links": [
    {
      "field": "specializes",
      "target": "R20TB2KEQQ"
    },
    {
      "field": "specializes",
      "target": "7CBT4SM83Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GHVW1Z35BS.md",
  "msc": [
    "57K32"
  ],
  "name": "Hyperbolic 3-manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GHV/GHVW1Z35BS.md",
  "status": null,
  "tag": "GHVW1Z35BS",
  "uri": "m2://GHVW1Z35BS",
  "url": "https://math2.tamnd.com/wiki/hyperbolic-3-manifold/",
  "writing": {
    "tag": "GHVW1Z35BS",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}