{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ be a closed manifold of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]].\nSuppose that $M$ carries a [[Riemannian metric|7CBT4SM83Y]] whose [[scalar curvature|ZHX2VYQDZ3]] is positive at every point.\n\n1. The [[Dirac operator|QMRAVX14BJ]] of $M$ for this metric has trivial kernel: every harmonic spinor is zero.\n2. If $n = 4k$, then the $\\hat{A}$-genus of $M$ vanishes: $\\hat{A}(M) = 0$.\n\n## Remarks\n\nLet $\\psi$ be a spinor with $D\\psi = 0$.\nIntegrating the [[Lichnerowicz formula|4WCJ49Z90S]] $D^2 = \\nabla^* \\nabla + \\frac{R}{4}$ against $\\psi$ gives $0 = \\int_M \\abs{\\nabla \\psi}^2 + \\frac{1}{4} \\int_M R \\abs{\\psi}^2$.\nBoth terms are nonnegative and $R \u003e 0$, so $\\psi = 0$.\nBy the Atiyah–Singer index theorem, $\\hat{A}(M)$ is the [[index of the positive part of the Dirac operator|JR2M58RCW3]], $\\dim \\ker D^+ - \\dim \\ker D^-$, which is then $0$.\nThe argument works as well when $R \\ge 0$ everywhere and $R \u003e 0$ at some point.\n\nExample: a K3 surface is a closed spin $4$-manifold with signature $-16$, so $\\hat{A} = -\\frac{\\sigma}{8} = 2 \\ne 0$.\nHence a K3 surface has no metric of positive scalar curvature.\n\nThe spin hypothesis is necessary.\nThe complex projective plane $\\C P^2$ with the Fubini–Study metric has positive scalar curvature, but $\\hat{A}(\\C P^2) = -\\frac{1}{8}$, computed from its Pontryagin number $p_1 = 3$ by $\\hat{A} = -\\frac{p_1}{24}$.\nThis is consistent because $\\C P^2$ is not spin.\n\n## Explanations\n\n### L1 human\n\nPositive scalar curvature is incompatible with certain topological invariants of spin manifolds.\nThe theorem shows that if the $\\hat{A}$-genus of a spin manifold is not zero, no metric on it can have positive scalar curvature everywhere.\nThe proof is short: positive curvature forces the Dirac operator to have no kernel, and the index theorem turns this into a statement about topology.\n\n### L2 human\n\nThis is the first and basic obstruction to metrics of positive scalar curvature.\nHitchin refined it to a $KO$-theoretic invariant.\nGromov and Lawson extended the method to Dirac operators twisted by bundles, which shows for example that tori admit no metric of positive scalar curvature.\nFor simply connected manifolds of dimension at least $5$, the refined invariant decides the question for spin manifolds.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "QMRAVX14BJ",
      "ZHX2VYQDZ3"
    ],
    "kind": "statement",
    "msc": [
      "53C27",
      "58J20",
      "53C21"
    ],
    "name": "Lichnerowicz vanishing theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence"
      }
    ],
    "tag": "CHJ4TAH0QX",
    "uses": [
      "QMRAVX14BJ",
      "4WCJ49Z90S",
      "RSPZ9TCFQ8",
      "ZHX2VYQDZ3",
      "7CBT4SM83Y",
      "JR2M58RCW3"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/CHJ4TAH0QX/",
  "links": [
    {
      "field": "about",
      "target": "QMRAVX14BJ"
    },
    {
      "field": "about",
      "target": "ZHX2VYQDZ3"
    },
    {
      "field": "uses",
      "target": "QMRAVX14BJ"
    },
    {
      "field": "uses",
      "target": "4WCJ49Z90S"
    },
    {
      "field": "uses",
      "target": "RSPZ9TCFQ8"
    },
    {
      "field": "uses",
      "target": "ZHX2VYQDZ3"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "JR2M58RCW3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CHJ4TAH0QX.md",
  "msc": [
    "53C27",
    "58J20",
    "53C21"
  ],
  "name": "Lichnerowicz vanishing theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CHJ/CHJ4TAH0QX.md",
  "status": null,
  "tag": "CHJ4TAH0QX",
  "uri": "m2://CHJ4TAH0QX",
  "url": "https://tamnd.github.io/math2/wiki/lichnerowicz-vanishing-theorem/",
  "writing": {
    "tag": "CHJ4TAH0QX",
    "kind": "statement",
    "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
      }
    ]
  }
}