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  "body": "\n## Summary\n\nAn isometry of Riemannian manifolds is a [[diffeomorphism|0KZZ70ZHXP]] that carries one [[Riemannian metric|Q61VFHCH2J]] to the other.\nTwo isometric manifolds are the same for Riemannian geometry: lengths, angles, geodesics and curvature correspond.\nThe isometries of a manifold onto itself form a group, which by the [[Myers–Steenrod theorem|Z3VRMW8ETQ]] is a Lie group.\n\n## Explanations\n\n### L1 human\n\nAn isometry moves a curved space onto another without stretching or shrinking anything, even infinitesimally.\nRotations of the round sphere are isometries of the sphere onto itself.\nRolling a flat sheet of paper into a cylinder is locally an isometry, because the paper is bent but not stretched.\nA map from the sphere to the plane can never be an isometry, even on a small piece, because the sphere is curved and the plane is flat.\n",
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