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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$, and let $U \\subset M$ be open.\nA *moving frame*, or local frame, on $U$ is an ordered $n$-tuple $(E_1, \\dots, E_n)$ of smooth vector fields on $U$.\nThe condition is that $(E_1(p), \\dots, E_n(p))$ is a basis of $T_pM$ for every $p \\in U$.\nThe *dual coframe* is the $n$-tuple of $1$-forms $(\\theta^1, \\dots, \\theta^n)$ on $U$ with $\\theta^i(E_j) = \\delta^i_j$.\n\n## Remarks\n\nA moving frame on $U$ is the same as a smooth section over $U$ of the [[frame bundle|F4HWTRNXAP]] of $M$.\nExample: on $\\R^2 \\setminus \\set{0}$ with polar coordinates, $E_1 = \\partial_r$ and $E_2 = \\frac1r \\partial_\\theta$ form a moving frame; at each point they are orthonormal for the Euclidean metric, so they form a basis.\nNon-example: on $\\R^2$, the pair $(\\partial_x, x \\, \\partial_y)$ is not a moving frame, because at the points with $x = 0$ the second field vanishes.\nGiven an affine connection $\\nabla$, the connection forms $\\omega^i_j$ of the frame are defined by $\\nabla E_j = \\sum_i \\omega^i_j \\otimes E_i$, and Cartan's structure equations express torsion and curvature through $d\\theta^i$ and $d\\omega^i_j$.\nA moving frame on all of $M$ exists only when the [[tangent bundle|WY6K3P4YCG]] is trivial; for example $S^2$ has none.\n",
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