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  "body": "\n## Problem\n\nDecide whether the sphere $S^6$ admits a complex structure.\nThat is, decide whether there is a [[complex manifold|JSGWAN2RJV]] of complex dimension $3$ whose underlying smooth manifold is diffeomorphic to the standard $S^6$.\nEquivalently, decide whether $S^6$ carries an integrable [[almost complex structure|9SYWRGVTRS]].\n\n## Remarks\n\nThe problem is open; several announced solutions have not been accepted.\nIt is often called the Hopf problem, after Heinz Hopf, who raised it in a paper published in 1948.\n\nWhat is known:\n\n1. $S^6$ has almost complex structures; one comes from identifying $S^6$ with the unit imaginary [[octonions|1HRMT3DSX2]] and using octonion multiplication.\n2. Borel and Serre showed that $S^2$ and $S^6$ are the only spheres with an almost complex structure; $S^2$ is the Riemann sphere, so $S^6$ is the only open case.\n3. The almost complex structure coming from the octonions is not integrable.\n4. LeBrun showed that no complex structure on $S^6$ is orthogonal with respect to the round metric.\n\nBy the [[Newlander–Nirenberg theorem|VCT848WV1T]], the problem asks for an almost complex structure whose Nijenhuis tensor vanishes.\n\n## Explanations\n\n### L1 human\n\nThe two-dimensional sphere is the Riemann sphere, the simplest compact complex manifold.\nThe question is whether the six-dimensional sphere can also be made into a complex manifold.\nIt is known to have the weaker structure of an almost complex manifold, which looks complex at each point separately.\nWhether this weaker structure can be made to fit together into true complex coordinates is still unknown.\n",
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  "front": {
    "about": [
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    "history": {
      "first_stated": "1948",
      "posed_by": "Heinz Hopf"
    },
    "kind": "problem",
    "msc": [
      "32Q60",
      "53C15"
    ],
    "name": "Complex structure on the six-sphere",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Almost_complex_manifold"
      },
      {
        "url": "https://arxiv.org/abs/1708.01068"
      }
    ],
    "tag": "V4M2HPJQPR",
    "uses": [
      "JSGWAN2RJV",
      "9SYWRGVTRS",
      "1HRMT3DSX2"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/V4M2HPJQPR/",
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  "markdown": "https://tamnd.github.io/math2/node/V4M2HPJQPR.md",
  "msc": [
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  "name": "Complex structure on the six-sphere",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/V4M/V4M2HPJQPR.md",
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  "tag": "V4M2HPJQPR",
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  "url": "https://tamnd.github.io/math2/wiki/complex-structure-on-the-six-sphere/",
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