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  "body": "\n## Statement\n\nLet $(M, g)$ and $(N, h)$ be connected [[Riemannian manifolds|7CBT4SM83Y]] with [[Riemannian distances|B4QR306D8G]] $d_g$ and $d_h$.\n\n1. If $\\varphi \\colon M \\to N$ is a surjective map with $d_h(\\varphi(p), \\varphi(q)) = d_g(p, q)$ for all $p, q \\in M$, then $\\varphi$ is smooth and is an [[isometry of Riemannian manifolds|R9YWQS15AA]].\n2. The group $\\operatorname{Isom}(M, g)$ of all isometries of $(M, g)$ onto itself, with the compact-open topology, is a Lie group, and its action on $M$ is smooth.\n3. If $M$ is [[compact|0TVJZYB6SH]], then $\\operatorname{Isom}(M, g)$ is compact.\n\n## Remarks\n\nThe idea of part 1 is that a distance-preserving map sends minimizing geodesics to minimizing geodesics, so in normal coordinates it acts as a linear isometry, and this forces smoothness.\nFor part 2, note that an isometry of a connected Riemannian manifold is determined by its value and its differential at one point.\nSo $\\operatorname{Isom}(M, g)$ embeds in the orthonormal frame bundle of $M$, and one shows that the image is a closed submanifold.\nFor the round sphere $S^n$, the theorem gives $\\operatorname{Isom}(S^n) = O(n + 1)$, a compact Lie group of dimension $n(n + 1)/2$.\nThe theorem is about Riemannian (positive definite) metrics; a pseudo-Riemannian metric defines no distance function.\n\n## Explanations\n\n### L1 human\n\nIf a map between two curved spaces keeps all distances, it automatically keeps the whole smooth geometry, with no further hypothesis.\nAlso, the symmetries of a curved space form a group that behaves like a group of matrices, with a finite number of independent directions of motion.\n\n### L2 human\n\nPart 1 lets one pass freely between the metric-space view of a Riemannian manifold and its smooth view.\nPart 2 is the starting point for the study of homogeneous and symmetric spaces.\nA Riemannian manifold whose isometry group acts transitively is $G/H$, with $G$ a Lie group and $H$ a compact subgroup.\nThe Lie algebra of $\\operatorname{Isom}(M, g)$ is identified with the complete [[Killing vector fields|5AKNNW3W4Z]] of $M$.\n",
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