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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$, and let $F(M) \\to M$ be its [[frame bundle|F4HWTRNXAP]].\nIt is the [[principal bundle|8E37E74XVP]] with group $\\mathrm{GL}(n, \\R)$ whose fibre over $p$ is the set of ordered bases of $T_pM$.\nLet $G$ be a Lie subgroup of $\\mathrm{GL}(n, \\R)$.\nA *$G$-structure* on $M$ is a principal $G$-subbundle $P \\subset F(M)$.\nThat is, $P$ is a smooth submanifold of $F(M)$ that meets every fibre and is preserved by the right action of $G$.\nMoreover $G$ acts freely and transitively on each fibre of $P$.\n\n## Remarks\n\nExample: a Riemannian metric $g$ gives an $\\mathrm{O}(n)$-structure, the bundle of $g$-orthonormal frames.\nTwo orthonormal frames at $p$ differ by an orthogonal matrix, so $\\mathrm{O}(n)$ acts simply transitively on each fibre.\nConversely every $\\mathrm{O}(n)$-structure is the orthonormal frame bundle of exactly one Riemannian metric.\nA $\\set{e}$-structure is a global frame, that is, a parallelization of $M$.\nNon-example: the sphere $S^2$ has no $\\set{e}$-structure, since a global frame would contain a nowhere vanishing vector field, which $S^2$ does not have.\nOther standard cases: $\\mathrm{GL}^+(n, \\R)$ gives orientations, $\\mathrm{GL}(m, \\C) \\subset \\mathrm{GL}(2m, \\R)$ gives almost complex structures, and $\\mathrm{Sp}(2m, \\R)$ gives nondegenerate $2$-forms.\nA $G$-structure is called integrable, or flat, when near each point there is a chart whose coordinate frames lie in $P$.\n",
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