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  "body": "\n## Summary\n\nA Riemannian metric on a smooth manifold is a smoothly varying [[inner product|0NB0AWGJBW]] on each tangent space.\nIt lets one measure lengths of tangent vectors and angles between them, and from these the lengths of curves, distances, areas and volumes.\nIt is a symmetric positive definite [[tensor field|WX302T93TW]] of type $(0, 2)$.\nEvery smooth manifold admits Riemannian metrics, and the choice of one is the starting point of Riemannian geometry.\n\n## Explanations\n\n### L1 human\n\nOn a curved surface, a small piece looks almost flat, and on a flat piece you can use a ruler and a protractor.\nA Riemannian metric gives such a ruler at every point, possibly different from point to point.\nTo measure a curve, add up the lengths of its tiny pieces with the local ruler.\nIn the upper half-plane, the metric $(dx^2 + dy^2)/y^2$ makes rulers shrink as you approach the $x$-axis, so the axis is infinitely far away.\nThis is the hyperbolic plane.\n",
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