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  "body": "\n## Summary\n\nThe Riemannian distance between two points of a connected [[Riemannian manifold|7CBT4SM83Y]] is the infimum of the lengths of the curves that join them.\nIt turns the manifold into a [[metric space|0VKD1TP7F3]] whose topology is the manifold topology.\nThis distance is the bridge between the infinitesimal data of the [[Riemannian metric|Q61VFHCH2J]] and global questions such as completeness, diameter and shortest paths.\n\n## Explanations\n\n### L1 human\n\nA Riemannian metric tells how long a tiny step is at each point.\nAdding up the tiny steps gives the length of a whole path.\nThe distance between two points is the length of the shortest way between them, or the limit of shorter and shorter ways when no shortest one exists.\nOn the round unit sphere, the distance between the north pole and a point on the equator is $\\pi/2$, the length of a quarter of a great circle.\nRemove one point from the plane, and take two points on opposite sides of the hole.\nTheir distance is the length of the straight segment, but no path in the punctured plane has this length.\n",
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