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  "body": "\n## Definition\n\nOn $\\R^7$ with coordinates $x_1, \\ldots, x_7$, write $dx_{ijk} = dx_i \\wedge dx_j \\wedge dx_k$ and let\n$$\\varphi_0 = dx_{123} + dx_{145} + dx_{167} + dx_{246} - dx_{257} - dx_{347} - dx_{356}.$$\nThe group $G_2$ is the group of linear maps $A \\in GL(7, \\R)$ with $A^* \\varphi_0 = \\varphi_0$.\nIt is a compact Lie group of dimension $14$ contained in $SO(7)$.\n\nLet $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $7$.\nIdentify the tangent space at a point with $\\R^7$ by a suitable linear isomorphism.\nThe manifold $M$ is a *$G_2$ manifold* if its [[holonomy|WHDZ3MK56B]] group is then contained in $G_2$.\n\n## Remarks\n\nEquivalently, $M$ carries a [[$3$-form|0E3G0GRPZ6]] $\\varphi$ that agrees with $\\varphi_0$ at each point in suitable linear coordinates, that determines $g$, and that is parallel for the [[Levi-Civita connection|CAZYGE1T4E]].\nSuch a $\\varphi$ is a [[$G_2$-structure|VYNQVZ8XVF]], and it is parallel exactly when $d\\varphi = 0$ and $d{*\\varphi} = 0$, where $*$ is the [[Hodge star|1Y4WX4XQWJ]] of $g$.\n\nExample: $\\R^7$ with the Euclidean metric and $\\varphi = \\varphi_0$; the form has constant coefficients, so it is parallel and the holonomy is trivial.\nThe flat torus $\\R^7 / \\Z^7$ is a compact example with the same form.\n\nNon-example: the round sphere $S^7$ is not a $G_2$ manifold.\nA metric with holonomy in $G_2$ is Ricci-flat, while the round metric has positive Ricci curvature.\n\nCompact manifolds with holonomy exactly $G_2$ were first constructed by Joyce, by resolving singularities of quotients of $T^7$ by finite groups.\n",
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