{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2DC88BMQVX"
    }
  ],
  "body": "\n## Example\n\nLet $G = \\mathrm{PSL}_2(\\R)$, acting on the upper half-plane $\\mathbb{H} = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$ by Möbius transformations $z \\mapsto (az + b)/(cz + d)$.\nThese maps are the orientation-preserving isometries of the hyperbolic metric $\\abs{dz}/\\operatorname{Im} z$.\nThe action of $G$ on the unit [[tangent bundle|DQB1BVQJWD]] $T^1\\mathbb{H}$ is free and transitive.\nLet $v_0$ be the upward unit vector at $i$; then $g \\mapsto g \\cdot v_0$ identifies $G$ with $T^1\\mathbb{H}$.\nFor $t \\in \\R$ let $a_t$ be the class of $\\begin{pmatrix} e^{t/2} \u0026 0 \\\\ 0 \u0026 e^{-t/2} \\end{pmatrix}$.\nThen $a_t \\cdot i = e^t i$, and the path $t \\mapsto e^t i$ is the unit-speed [[geodesic|NSYMQMHPDN]] through $v_0$, since the hyperbolic distance from $i$ to $e^t i$ is $t$ for $t \\ge 0$.\nAs $g$ is an isometry, $g a_t \\cdot v_0$ is the unit vector reached after time $t$ along the geodesic starting at $g \\cdot v_0$.\nSo the geodesic flow on $T^1\\mathbb{H}$ is the right multiplication $g \\mapsto g a_t$.\n\nNow let $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$, for example the [[modular group|VEKV1EM77B]] or a torsion-free cocompact lattice.\nLeft multiplication by $\\Gamma$ commutes with right multiplication by $a_t$.\nSo the flow descends to $\\Gamma \\backslash G$, which is the unit tangent bundle of the locally symmetric surface $\\Gamma \\backslash \\mathbb{H}$ when $\\Gamma$ is torsion-free.\nIt is an action of the Lie group $\\R$ on $\\Gamma \\backslash G$.\nThe same recipe works for a quotient $\\Gamma \\backslash G / K$ of a Riemannian symmetric space $G/K$.\nThere geodesics are orbits of one-parameter subgroups, and the flow becomes right multiplication on a homogeneous space of $G$.\n\n## Explanations\n\n### L1 human\n\nOn a curved surface, a geodesic flow moves every direction arrow forward along its straightest path at unit speed.\nOn a hyperbolic surface built from the group $\\mathrm{PSL}_2(\\R)$, this complicated-looking motion is just multiplication on the right by a diagonal matrix.\nThat turns a question in geometry into a question about groups.\nFacts about group actions, such as their mixing behaviour, then say how geodesics wander over the surface.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "illustrates": [
      "NSYMQMHPDN"
    ],
    "instance_of": [
      "96B4E2GPKJ"
    ],
    "kind": "example",
    "msc": [
      "37D40",
      "22E40",
      "53C35"
    ],
    "name": "Geodesic flow on a locally symmetric space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Geodesic"
      }
    ],
    "tag": "5TKBSHY6VC",
    "uses": [
      "BTYF5Y5PFY",
      "DQB1BVQJWD",
      "VEKV1EM77B"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/5TKBSHY6VC/",
  "links": [
    {
      "field": "instance_of",
      "target": "96B4E2GPKJ"
    },
    {
      "field": "illustrates",
      "target": "NSYMQMHPDN"
    },
    {
      "field": "uses",
      "target": "BTYF5Y5PFY"
    },
    {
      "field": "uses",
      "target": "DQB1BVQJWD"
    },
    {
      "field": "uses",
      "target": "VEKV1EM77B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5TKBSHY6VC.md",
  "msc": [
    "37D40",
    "22E40",
    "53C35"
  ],
  "name": "Geodesic flow on a locally symmetric space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5TK/5TKBSHY6VC.md",
  "status": null,
  "tag": "5TKBSHY6VC",
  "uri": "m2://5TKBSHY6VC",
  "url": "https://tamnd.github.io/math2/wiki/geodesic-flow-on-a-locally-symmetric-space/",
  "writing": {
    "tag": "5TKBSHY6VC",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}