{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "GG6NYPKMHX"
    }
  ],
  "body": "\n## Statement\n\nLet $S \\subset \\R^3$ be an [[oriented|0TD904MG06]] [[regular surface|6080X3FAD4]], and let $R \\subset S$ be a compact region.\nAssume that the boundary $\\partial R$ is a finite union of simple closed, piecewise regular curves $C_1, \\ldots, C_m$, positively oriented.\nLet $\\theta_1, \\ldots, \\theta_p$ be the external angles at the vertices of these curves.\nLet $K$ be the [[Gaussian curvature|SQVQ5DQT6W]] of $S$, and let $k_g$ be the [[geodesic curvature|550P8PFMVC]] of the curves $C_i$.\nLet $\\chi(R)$ be the [[Euler characteristic|RNDDWDAFJ5]] of $R$.\nThen\n$$\\int_R K \\, dA + \\sum_{i=1}^{m} \\int_{C_i} k_g \\, ds + \\sum_{l=1}^{p} \\theta_l = 2\\pi \\chi(R).$$\nIn particular, if $S$ is compact, then\n$$\\int_S K \\, dA = 2\\pi \\chi(S).$$\n\n## Remarks\n\nThe proof starts from the local case of a small simple region in one coordinate patch.\nThere the formula follows from Green's theorem, applied to the turning of a tangent vector along the boundary.\nOne then takes a [[triangulation|YH170WBQ6M]] of $R$ fine enough that each triangle lies in one patch, and adds the local formulas.\nInterior edges are traversed twice in opposite directions, so their geodesic curvature terms cancel.\nThe angle terms then add up to $2\\pi(V - E + F) = 2\\pi\\chi(R)$.\n\nFor the round sphere of radius $r$, one has $K = 1/r^2$ and area $4\\pi r^2$, so the total curvature is $4\\pi = 2\\pi \\cdot 2$, and indeed $\\chi(S^2) = 2$.\nFor any torus in $\\R^3$, the total curvature is $0$, since $\\chi = 0$.\nCompactness is needed for the second formula: the plane has $K = 0$ but $\\chi = 1$.\n\nThe theorem extends to compact two-dimensional Riemannian manifolds, and the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]] extends it to all even dimensions.\n\n## Explanations\n\n### L1 human\n\nGaussian curvature measures how a surface bends at each point.\nThe theorem says that if you add up all the curvature over a closed surface, you get $2\\pi$ times a whole number that depends only on the topological shape.\nSo a sphere, however you dent it, always has total curvature $4\\pi$.\nA local, metric quantity is tied to a global, topological one.\n\n### L2 human\n\nThe formula is the standard bridge between the local geometry and the topology of surfaces.\nIt shows that a compact surface with $K \u003e 0$ everywhere must have $\\chi \u003e 0$, so it is a sphere when orientable.\nApplied to a geodesic triangle, where $k_g = 0$, it gives $\\int_T K \\, dA = \\alpha + \\beta + \\gamma - \\pi$ for the interior angles $\\alpha, \\beta, \\gamma$.\nThis explains why angle sums exceed $\\pi$ on a sphere and fall short of $\\pi$ on surfaces of negative curvature.\nIt is also the first case of the theory of characteristic classes, where the Euler class is represented by a curvature form.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "SQVQ5DQT6W",
      "RNDDWDAFJ5"
    ],
    "kind": "statement",
    "msc": [
      "53A05",
      "53C45",
      "57M50"
    ],
    "name": "Gauss–Bonnet theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q742833",
      "wikipedia": "Gauss–Bonnet theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem"
      }
    ],
    "tag": "ESGEHKPD60",
    "uses": [
      "6080X3FAD4",
      "SQVQ5DQT6W",
      "550P8PFMVC",
      "RNDDWDAFJ5",
      "0TD904MG06"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/ESGEHKPD60/",
  "links": [
    {
      "field": "about",
      "target": "SQVQ5DQT6W"
    },
    {
      "field": "about",
      "target": "RNDDWDAFJ5"
    },
    {
      "field": "uses",
      "target": "6080X3FAD4"
    },
    {
      "field": "uses",
      "target": "SQVQ5DQT6W"
    },
    {
      "field": "uses",
      "target": "550P8PFMVC"
    },
    {
      "field": "uses",
      "target": "RNDDWDAFJ5"
    },
    {
      "field": "uses",
      "target": "0TD904MG06"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ESGEHKPD60.md",
  "msc": [
    "53A05",
    "53C45",
    "57M50"
  ],
  "name": "Gauss–Bonnet theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ESG/ESGEHKPD60.md",
  "status": null,
  "tag": "ESGEHKPD60",
  "uri": "m2://ESGEHKPD60",
  "url": "https://math2.tamnd.com/wiki/gauss-bonnet-theorem/",
  "writing": {
    "tag": "ESGEHKPD60",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}