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  "body": "\n## Problem\n\nLet $M$ be a connected smooth manifold, and let $\\Delta \\subseteq TM$ be a smooth subbundle of the tangent bundle, the horizontal distribution.\nLet $g$ be a smooth inner product on the fibres of $\\Delta$, so that $(M, \\Delta, g)$ is a [[sub-Riemannian manifold|97NYHNBZ2D]].\nA curve $\\gamma \\colon [0, 1] \\to M$ is *horizontal* if it is [[absolutely continuous|ZT6F4C2YHK]] and $\\gamma'(t) \\in \\Delta_{\\gamma(t)}$ for almost every $t$.\nIts length is $\\ell(\\gamma) = \\int_0^1 \\sqrt{g(\\gamma'(t), \\gamma'(t))} \\, dt$.\nLet $a, b \\in M$.\nThe *sub-Riemannian geodesic problem* is to find a horizontal curve from $a$ to $b$ of least length among all horizontal curves from $a$ to $b$.\n\n## Remarks\n\nIf $\\Delta$ is bracket generating, the [[Chow–Rashevskii theorem|VQPN24VDH1]] shows that any two points are joined by a horizontal curve.\nSo the infimum of lengths, the sub-Riemannian distance, is finite.\nIf moreover the distance is complete, any two points are joined by a length minimizer, by a compactness argument.\nLocally, with an orthonormal frame $X_1, \\dots, X_k$ of $\\Delta$, the problem is the [[optimal control|T3V6ZNAMDN]] problem $\\gamma' = \\sum_{i=1}^k u_i X_i(\\gamma)$ with cost $\\int_0^1 \\sum_i u_i^2 \\, dt$.\n[[Pontryagin's maximum principle|7RPV2T29CF]] gives normal geodesics, which are projections of solutions of a Hamiltonian system on $T^*M$, and abnormal extremals.\nUnlike in Riemannian geometry, some length minimizers can be abnormal and fail to be normal geodesics.\nExample: in the Heisenberg group, $\\R^3$ with $\\Delta$ spanned by $\\partial_x - \\frac{y}{2} \\partial_z$ and $\\partial_y + \\frac{x}{2} \\partial_z$, the geodesics project to lines and circles in the $(x, y)$-plane.\nThe length of a horizontal curve is the length of its projection.\nTake a horizontal curve that starts at the origin.\nIts change in $z$ equals the signed area enclosed by its projection and the segment from the endpoint back to the origin.\nSo the problem becomes a version of Dido's isoperimetric problem.\n\n## Explanations\n\n### L1 human\n\nIn a sub-Riemannian space one may only move in certain allowed directions at each point, like a car that cannot slide sideways.\nStill, by combining allowed moves, one can reach every point, just as a car can parallel park.\nThe geodesic problem asks for the shortest allowed path between two points.\nThese shortest paths can look very different from straight lines, and they connect geometry with optimal control.\n",
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      "first_stated": "1938",
      "notes": "Rashevskii (1938) and Chow (1939) proved that a bracket-generating distribution joins any two points by a horizontal curve, so that the shortest such curve is defined.",
      "posed_by": "Petr Rashevskii and Wei-Liang Chow"
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