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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$ with [[tangent bundle|WY6K3P4YCG]] $TM$.\nA *frame* at $p \\in M$ is an ordered [[basis|0KCQ9Y3QKA]] $e = (e_1, \\ldots, e_n)$ of the tangent space $T_pM$.\nThe *frame bundle* of $M$ is the set\n$$F(M) = \\set{(p, e) : p \\in M, \\ e \\text{ a frame at } p}.$$\nIt comes with the projection $\\pi(p, e) = p$ and the right action of the [[general linear group|0FCHE902D7]] $GL(n, \\R)$ given by\n$$(p, e) \\cdot g = (p, e g), \\qquad (e g)_j = \\sum_{i=1}^{n} g_{ij} \\, e_i.$$\nFor a chart $(U, x)$ of $M$ and a frame $e$ at $p \\in U$, let $A$ be the matrix with $e_j = \\sum_i A_{ij} \\, \\partial/\\partial x^i|_p$.\n$F(M)$ carries the smooth structure for which each map $(p, e) \\mapsto (p, A)$ is a diffeomorphism $\\pi^{-1}(U) \\to U \\times GL(n, \\R)$.\nWith this structure $F(M)$ is a [[principal|8E37E74XVP]] $GL(n, \\R)$-bundle over $M$.\n\n## Remarks\n\nFor $M = \\R^n$, a frame at $p$ is the same as an invertible matrix, so $F(\\R^n) = \\R^n \\times GL(n, \\R)$.\nCheck of the action: $(e g) h = e (g h)$, because $((eg)h)_k = \\sum_j h_{jk} \\sum_i g_{ij} e_i = \\sum_i (gh)_{ik} e_i$.\n\nA tuple $(e_1, \\ldots, e_n)$ of tangent vectors at $p$ that is linearly dependent, such as $e_1 = e_2$, is not a frame and is not a point of $F(M)$.\n\nIf $M$ is a Riemannian manifold, the orthonormal frames form the orthonormal frame bundle, a principal $O(n)$-bundle.\nThe same construction applies to any vector bundle $E$ of rank $k$ and gives a principal $GL(k, \\R)$-bundle.\n\n## Explanations\n\n### L1 human\n\nThe frame bundle of a manifold is the set of all pairs made of a point and an ordered basis of the tangent space at that point.\n",
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