{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *hyperbolic plane* is a model of the axioms of [[absolute geometry|VKFPCSRNDH]], including Dedekind's axiom of continuity, in which the hyperbolic parallel axiom holds.\nThat axiom says: for each line $\\ell$ and each point $P$ not on $\\ell$, at least two distinct lines through $P$ do not meet $\\ell$.\n*Hyperbolic geometry* is the study of the hyperbolic plane and of its higher-dimensional analogues.\n\n## Remarks\n\nAny two hyperbolic planes in this sense are isomorphic, so hyperbolic geometry is a [[non-Euclidean geometry|SPCYMFN5V2]] with a single model up to isomorphism.\nIn differential geometry, the hyperbolic $n$-space $\\mathbb{H}^n$ is the complete simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with constant sectional curvature $-1$.\nFor $n = 2$ this is a surface of [[Gaussian curvature|SQVQ5DQT6W]] $-1$, and it is a hyperbolic plane in the axiomatic sense.\nIn it, a triangle with angles $\\alpha, \\beta, \\gamma$ has area $\\pi - \\alpha - \\beta - \\gamma$.\n\nExample: the [[Poincaré half-plane model|RHKSMPBSQ1]] $\\set{(x, y) : y \u003e 0}$ with the metric $(dx^2 + dy^2)/y^2$ satisfies the axioms.\nIts lines are the vertical half-lines and the semicircles centred on the $x$-axis.\n\nNon-example: the Euclidean plane satisfies the axioms of absolute geometry.\nIt fails the hyperbolic parallel axiom, because each point off a line lies on exactly one parallel.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GN4N94GQ0Z",
    "kind": "definition",
    "msc": [
      "51M10",
      "53A35"
    ],
    "name": "Hyperbolic geometry",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyperbolic_geometry"
      }
    ],
    "tag": "RBZPZTSPZE",
    "uses": [
      "SPCYMFN5V2",
      "VKFPCSRNDH",
      "7CBT4SM83Y",
      "SQVQ5DQT6W"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/RBZPZTSPZE/",
  "links": [
    {
      "field": "defines",
      "target": "GN4N94GQ0Z"
    },
    {
      "field": "uses",
      "target": "SPCYMFN5V2"
    },
    {
      "field": "uses",
      "target": "VKFPCSRNDH"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "SQVQ5DQT6W"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RBZPZTSPZE.md",
  "msc": [
    "51M10",
    "53A35"
  ],
  "name": "Hyperbolic geometry",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RBZ/RBZPZTSPZE.md",
  "status": null,
  "tag": "RBZPZTSPZE",
  "uri": "m2://RBZPZTSPZE",
  "url": "https://tamnd.github.io/math2/wiki/hyperbolic-geometry-definition/",
  "writing": {
    "tag": "RBZPZTSPZE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}