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  "body": "\n## Definition\n\nLet $\\varphi_0$ be the standard $3$-form of $G_2$ on $\\R^7$, and let $*_7$ be the Hodge star of $\\R^7$.\nThe Cayley $4$-form on $\\R^8 = \\R \\oplus \\R^7$ is $\\Phi_0 = dx_0 \\wedge \\varphi_0 + *_7 \\varphi_0$.\nThe group $\\operatorname{Spin}(7)$ is identified with the group of linear maps $A \\in GL(8, \\R)$ with $A^* \\Phi_0 = \\Phi_0$.\nIt is a compact Lie group of dimension $21$ contained in $SO(8)$.\n\nLet $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $8$.\nIdentify the tangent space at a point with $\\R^8$ by a suitable linear isomorphism.\nThe manifold $M$ is a *Spin(7)-manifold* if its [[holonomy|WHDZ3MK56B]] group is then contained in $\\operatorname{Spin}(7)$.\n\n## Remarks\n\nEquivalently, $M$ carries a [[$4$-form|0E3G0GRPZ6]] $\\Phi$ that agrees with $\\Phi_0$ at each point in suitable linear coordinates and is parallel for the [[Levi-Civita connection|CAZYGE1T4E]]; such a $\\Phi$ determines $g$.\n\nExample: $\\R^8$ with $\\Phi = \\Phi_0$, whose constant coefficients make it parallel.\nIf $(N, \\varphi)$ is a [[$G_2$ manifold|P1TFW9QWAD]], then $N \\times S^1$ with $\\Phi = dt \\wedge \\varphi + *\\varphi$ is a Spin(7)-manifold, with holonomy contained in $G_2$.\n\nNon-example: the round sphere $S^8$ is not a Spin(7)-manifold, since a metric with holonomy in $\\operatorname{Spin}(7)$ is Ricci-flat and the round metric is not.\n\nCompact manifolds with holonomy exactly $\\operatorname{Spin}(7)$ were first constructed by Joyce.\n",
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