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  "body": "\n## Summary\n\nA sub-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] where motion is allowed only along chosen horizontal directions.\nThese form a subbundle of the tangent bundle, and they carry an inner product.\nDistance is the infimum of lengths of curves that are tangent to the horizontal directions at every time.\nAssume that the horizontal directions and their [[Lie brackets|68FGQEBD6J]] span all directions.\nThen, by the Chow–Rashevskii theorem, any two points of a connected manifold are joined by a horizontal curve, and the distance is finite.\nSub-Riemannian geometry appears in control theory, in the study of hypoelliptic operators and in the geometry of the Heisenberg group.\n\n## Explanations\n\n### L1 human\n\nThink of a car: it can drive forward and backward and turn its wheels, but it cannot move sideways.\nStill, by combining these moves, as in parallel parking, it can reach every position.\nA sub-Riemannian manifold models this: only some directions of motion are allowed, but combining them reaches everywhere.\nThe distance between two places is then the length of the shortest allowed route, which can be much longer than in the unrestricted space.\n",
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