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  "body": "\n## Summary\n\nA $G$-structure on an $n$-manifold is a choice, at every point, of a preferred family of frames of the tangent space.\nAny two preferred frames at a point are related by an element of a fixed subgroup $G$ of $\\mathrm{GL}(n, \\R)$.\nFormally it is a reduction of the [[frame bundle|F4HWTRNXAP]] to the group $G$.\nMany geometric structures are $G$-structures: Riemannian metrics for $G = \\mathrm{O}(n)$, orientations for $\\mathrm{GL}^+(n, \\R)$, almost complex structures for $\\mathrm{GL}(m, \\C)$ with $n = 2m$.\nThe notion gives one language for comparing and classifying these structures.\n\n## Explanations\n\n### L1 human\n\nAt each point of a manifold one can choose many bases of the tangent space.\nA geometric structure picks out some of them as the good ones.\nA Riemannian metric picks out the orthonormal bases, and any two of these differ by a rotation or reflection, an element of $\\mathrm{O}(n)$.\nAn orientation picks out the positively oriented bases, which differ by matrices of positive determinant.\nA $G$-structure is this idea for an arbitrary group $G$.\n",
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