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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete noncompact [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[sectional curvature|1X42JAA349]].\nThen $M$ contains a compact, totally convex, totally geodesic embedded submanifold $S$ without boundary, called a *soul* of $M$.\nMoreover, $M$ is [[diffeomorphic|0KZZ70ZHXP]] to the total space of the normal vector bundle of $S$ in $M$.\n\n## Remarks\n\nThe theorem is due to Cheeger and Gromoll (1972).\nThe idea of the proof: fix a point $p$ and remove from $M$ the open balls $B(\\gamma(t), t)$ for every ray $\\gamma$ from $p$ and every $t \u003e 0$.\nThe remaining set is compact, and it is totally convex by the Toponogov comparison theorem.\nShrinking this set step by step through levels of the distance to its boundary ends at a totally convex set without boundary, the soul.\nA deformation along the gradient-like flow of the distance to $S$ gives the diffeomorphism with the normal bundle.\n\nExample: the flat cylinder $S^1 \\times \\R$ has the soul $S^1 \\times \\set{0}$, and it is the normal bundle of this circle.\nThe paraboloid $z = x^2 + y^2$ has positive curvature, its soul is a single point, and it is diffeomorphic to $\\R^2$.\n\nA soul is not unique as a subset, but any two souls are isometric.\nPerelman proved in 1994 the soul conjecture of Cheeger and Gromoll.\nIt says: if moreover the sectional curvature is positive at some point, then the soul is a point and $M$ is diffeomorphic to $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nA complete, infinite space that is nowhere negatively curved has a compact core, the soul.\nThe whole space is this core with directions going off to infinity attached, like a cylinder around its central circle.\nSo the topology of the space is that of the soul.\n\n### L2 human\n\nThe soul theorem reduces the topology of open manifolds with $K \\ge 0$ to that of compact ones: $M$ is homotopy equivalent to its soul.\nIt is a main example of how Toponogov's comparison theorem is used to build convex sets.\n",
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