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  "body": "\n## Statement\n\nLet $M$ be a closed smooth manifold of odd dimension.\nLet $\\alpha$ be a contact form on $M$, so that $(M, \\ker \\alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].\nThen the [[Reeb vector field|GGM63KXCF2]] $R_\\alpha$ of $\\alpha$ has at least one [[periodic orbit|0AA97XA70K]].\nThat is, there are a point $x \\in M$ and a time $T \u003e 0$ such that the flow of $R_\\alpha$ maps $x$ to itself at time $T$.\n\n## Remarks\n\nWeinstein stated the conjecture for hypersurfaces of contact type in symplectic manifolds.\nHe was motivated by Rabinowitz's theorem on periodic orbits on star-shaped energy surfaces in $\\R^{2n}$.\nThe form above is the standard intrinsic version.\nViterbo proved it for hypersurfaces of contact type in $\\R^{2n}$ in 1987.\nHofer proved it for $S^3$ and for closed contact $3$-manifolds that are overtwisted or have $\\pi_2 \\ne 0$, using pseudoholomorphic curves in the symplectization.\nTaubes proved it for all closed $3$-manifolds in 2007, using Seiberg–Witten Floer homology.\nIn dimension $5$ and higher it is open in general, although it is known in many cases.\n\n## Explanations\n\n### L1 human\n\nOn a closed contact manifold, a contact form determines a flow, the Reeb flow.\nThe conjecture says that this flow always has a closed orbit, a trajectory that comes back to where it started.\nFor energy surfaces of a mechanical system, this means a periodic motion with that energy.\n",
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      "first_stated": "1979",
      "posed_by": "Alan Weinstein"
    },
    "kind": "conjecture",
    "msc": [
      "53D10",
      "37J55",
      "53D42"
    ],
    "name": "Weinstein conjecture",
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      "wikidata": "Q966711",
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        "url": "https://en.wikipedia.org/wiki/Weinstein_conjecture"
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    "tag": "MP6B64DS31",
    "uses": [
      "GGM63KXCF2",
      "HBQ8BZ7XB5",
      "0AA97XA70K"
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/MP6B64DS31/",
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  "msc": [
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  "name": "Weinstein conjecture",
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    "tag": "MP6B64DS31",
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