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  "body": "\n## Statement\n\nLet $S$ and $\\bar S$ be [[regular surfaces|6080X3FAD4]] in $\\R^3$, and let $\\varphi \\colon S \\to \\bar S$ be a smooth map.\nAssume that $\\varphi$ is a local isometry: for each $p \\in S$ and $v, w \\in T_p S$, $\\ip{d\\varphi_p(v)}{d\\varphi_p(w)} = \\ip{v}{w}$.\nLet $K$ and $\\bar K$ be the [[Gaussian curvatures|SQVQ5DQT6W]] of $S$ and $\\bar S$.\nThen $\\bar K(\\varphi(p)) = K(p)$ for each $p \\in S$.\n\n## Remarks\n\nThe key step is that $K$ can be written, in any local parametrisation, through $E, F, G$ of the [[first fundamental form|MAJ3DH7MZS]] and their partial derivatives of order at most $2$.\nThis follows from the Gauss equation among the [[Gauss–Codazzi equations|68TVVHCQ3Z]], which expresses $eg - f^2$ through the Christoffel symbols, and these depend only on $E, F, G$.\nA local isometry has, in suitable parametrisations, the same functions $E, F, G$, hence the same $K$.\nExample: the map $(u, v) \\mapsto (\\cos u, \\sin u, v)$ wraps the plane onto the unit cylinder as a local isometry, and indeed both have $K = 0$.\nConsequence: no open piece of a sphere, where $K \u003e 0$, is isometric to a piece of the plane, so every flat map of a region of the Earth distorts some distances.\nThe converse fails: a map that preserves $K$ need not be a local isometry.\nFor example, the map $(x, y, 0) \\mapsto (2x, y, 0)$ of the plane keeps $K = 0$ but doubles some lengths.\n\n## Explanations\n\n### L1 human\n\nGaussian curvature can be measured by someone who lives on the surface and only measures lengths on it.\nBending a surface without stretching it, like rolling a sheet of paper, never changes its Gaussian curvature.\nThis is why you cannot flatten an orange peel without tearing it, and why every flat map of the Earth is distorted.\n\n### L2 human\n\nGauss called this result remarkable because $K$ was defined through the normal and the second fundamental form, which depend on how the surface sits in space.\nThe theorem shows that $K$ is intrinsic, and it is the starting point of Riemannian geometry, where curvature is defined from the metric alone.\nIn orthogonal coordinates, where $F = 0$, one has\n$$K = -\\frac{1}{2\\sqrt{EG}} \\left( \\frac{\\partial}{\\partial u} \\frac{G_u}{\\sqrt{EG}} + \\frac{\\partial}{\\partial v} \\frac{E_v}{\\sqrt{EG}} \\right).$$\nThe mean curvature is not intrinsic: the plane and the cylinder are locally isometric but have $H = 0$ and $H \\ne 0$.\n",
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