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  "body": "\n## Definition\n\nLet $M$ be a connected [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$.\nA *sub-Riemannian manifold* is a triple $(M, H, g)$ such that:\n\n1. $H \\subseteq TM$ is a smooth subbundle, the horizontal distribution;\n2. $H$ is bracket-generating: at each $p \\in M$, the vector fields tangent to $H$ and their iterated [[Lie brackets|68FGQEBD6J]] span $T_pM$;\n3. $g$ assigns to each $p \\in M$ an [[inner product|0NB0AWGJBW]] $g_p$ on the fibre $H_p$, depending smoothly on $p$.\n\nA horizontal curve is an absolutely continuous curve $\\gamma$ with $\\gamma'(t) \\in H_{\\gamma(t)}$ for almost every $t$.\nThe sub-Riemannian distance $d(p, q)$ is the infimum of $\\int \\sqrt{g(\\gamma', \\gamma')} \\, dt$ over the horizontal curves from $p$ to $q$.\n\n## Remarks\n\nExample: the Heisenberg group is $\\R^3$ with $H$ spanned by $X = \\partial_x - \\tfrac{y}{2} \\partial_z$ and $Y = \\partial_y + \\tfrac{x}{2} \\partial_z$, with $X, Y$ orthonormal.\nHere $[X, Y] = \\partial_z$, so $X, Y, [X, Y]$ span $T_p\\R^3$ at each point, and clause 2 holds.\n\nNon-example: on $\\R^3$, let $H$ be spanned by $\\partial_x$ and $\\partial_y$.\nAll brackets of fields tangent to $H$ stay in $H$, so clause 2 fails; horizontal curves stay in the planes $z = \\text{const}$, and points with different $z$ are at infinite distance.\n\nWhen $H = TM$, the definition gives a Riemannian manifold.\nSome authors do not require clause 2 in the definition and add it as a hypothesis when needed.\n",
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