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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[Ricci curvature|4GTNM0WHDH]].\nSuppose $M$ contains a line, that is, a unit-speed [[geodesic|NSYMQMHPDN]] $\\gamma \\colon \\R \\to M$ with $d(\\gamma(s), \\gamma(t)) = \\abs{s - t}$ for all $s, t \\in \\R$, where $d$ is the [[Riemannian distance|B4QR306D8G]].\nThen $M$ is [[isometric|R9YWQS15AA]] to a Riemannian product $N \\times \\R$, where $N$ is a connected complete Riemannian manifold with nonnegative Ricci curvature.\n\n## Remarks\n\nThe idea of the proof: let $b^{+}$ and $b^{-}$ be the Busemann functions of the two rays $t \\mapsto \\gamma(t)$ and $t \\mapsto \\gamma(-t)$, $t \\ge 0$.\nThe Laplacian comparison theorem gives $\\Delta b^{\\pm} \\le 0$ in a weak sense.\nThe triangle inequality gives $b^{+} + b^{-} \\ge 0$, with equality on $\\gamma$, so the strong maximum principle forces $b^{+} + b^{-} = 0$ and $b^{+}$ is harmonic and smooth.\nThe [[Bochner formula|DF7KYMAGTC]] applied to $b^{+}$, with $\\abs{\\nabla b^{+}} = 1$, gives that its Hessian vanishes, so $\\nabla b^{+}$ is a parallel vector field and $M$ splits along its flow.\n\nExample: the flat cylinder $S^1 \\times \\R$ contains the lines $\\set{\\theta} \\times \\R$ and splits with $N = S^1$.\n\nThe curvature hypothesis is needed: the hyperbolic plane contains lines but is not a product with a line.\nThe line is needed too: the paraboloid $z = x^2 + y^2$ has positive curvature and contains no line.\n\n## Explanations\n\n### L1 human\n\nSuppose a space is nowhere negatively curved in the Ricci sense and contains one infinitely long shortest path.\nThen the whole space is a product of a cross-section and that line, like a cylinder.\n\n### L2 human\n\nToponogov proved the theorem earlier under the stronger hypothesis of nonnegative sectional curvature; Cheeger and Gromoll proved the Ricci version in 1971.\nA main consequence: if $M$ is compact with $\\operatorname{Ric} \\ge 0$, then its universal cover is isometric to $N \\times \\R^k$ with $N$ compact.\nIt follows that the fundamental group of such $M$ contains a free abelian subgroup of finite index.\nVersions of the theorem for limit spaces are central in the work of Cheeger and Colding.\n",
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  "name": "Cheeger–Gromoll splitting theorem",
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