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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \\ge 2$ whose [[sectional curvature|1X42JAA349]] satisfies $K \\ge k$ for a real number $k$.\nLet $M_k^2$ be the simply connected complete surface of constant curvature $k$, a [[space form|1VM1AQT956]].\nLet $p \\in M$, and let $\\gamma_1, \\gamma_2$ be minimizing unit-speed [[geodesics|NSYMQMHPDN]] starting at $p$, of lengths $\\ell_1, \\ell_2$, that make the angle $\\alpha \\in [0, \\pi]$ at $p$.\nIf $k \u003e 0$, assume $\\ell_1, \\ell_2 \\le \\pi / \\sqrt{k}$.\nLet $\\bar\\gamma_1, \\bar\\gamma_2$ be unit-speed geodesics in $M_k^2$ starting at a point $\\bar p$, of lengths $\\ell_1, \\ell_2$, that make the angle $\\alpha$ at $\\bar p$.\nThen\n$$d\\bigl(\\gamma_1(\\ell_1), \\gamma_2(\\ell_2)\\bigr) \\le \\bar d\\bigl(\\bar\\gamma_1(\\ell_1), \\bar\\gamma_2(\\ell_2)\\bigr),$$\nwhere $d$ is the [[Riemannian distance|B4QR306D8G]] of $M$ and $\\bar d$ that of $M_k^2$.\n\n## Remarks\n\nThis is the hinge form of the theorem.\nAn equivalent form compares the angles of a geodesic triangle with minimizing sides in $M$ with those of the triangle with the same side lengths in $M_k^2$.\nEach angle in $M$ is at least the corresponding angle in $M_k^2$.\nThe proof starts from the Rauch comparison theorem, which gives the result for short hinges, and extends it to all hinges by a subdivision argument that uses completeness.\n\nExample: if $K \\ge 0$, compare with the Euclidean plane.\nThen for two minimizing geodesics of lengths $\\ell_1, \\ell_2$ from $p$ at angle $\\alpha$, the endpoints are at distance at most $\\sqrt{\\ell_1^2 + \\ell_2^2 - 2 \\ell_1 \\ell_2 \\cos\\alpha}$, the value given by the law of cosines.\nOn the round sphere, for $0 \u003c \\alpha \u003c \\pi$, the distance is strictly smaller, since triangles on a sphere are fatter than flat triangles.\n\n## Explanations\n\n### L1 human\n\nOn a space that curves at least as much as a model surface, triangles are \"fatter\" than in the model.\nTwo paths that leave a point at a given angle end up closer together than they would in the model.\n\n### L2 human\n\nToponogov's theorem turns a lower curvature bound into a statement about distances only, without derivatives.\nThis is why it is the basis of many global results such as the soul theorem and the Grove–Shiohama diameter sphere theorem.\nIts conclusion is also used as the definition of a lower curvature bound for metric spaces, which leads to Alexandrov spaces.\n",
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