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G2 manifold
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{{Short description|Seven-dimensional Riemannian manifold}} {{DISPLAYTITLE:G<sub>2</sub> manifold}} {{CS1 config|mode=cs2}} In [[differential geometry]], a '''''G''<sub>2</sub> manifold''' or '''Joyce manifold''' is a seven-dimensional [[Riemannian manifold]] with [[holonomy group]] contained in [[G2 (mathematics)|''G''<sub>2</sub>]]. The [[group (mathematics)|group]] <math>G_2</math> is one of the five exceptional [[simple Lie group]]s. It can be described as the [[automorphism group]] of the [[octonion]]s, or equivalently, as a proper subgroup of [[special orthogonal group]] SO(7) that preserves a [[spinor]] in the eight-dimensional [[spinor representation]] or lastly as the subgroup of the [[general linear group]] GL(7) which preserves the non-degenerate 3-form <math>\phi</math>, the associative form. The [[Hodge dual]], <math>\psi=*\phi</math> is then a parallel 4-form, the coassociative form. These forms are [[calibrated geometry|calibrations]] in the sense of Reese Harvey and [[H. Blaine Lawson]],<ref>{{citation | first1 = Reese | last1 = Harvey | first2 = H. Blaine | last2 = Lawson|author-link2=H. Blaine Lawson | title = Calibrated geometries | journal = [[Acta Mathematica]] | volume = 148 | year = 1982 | pages = 47–157 | doi=10.1007/BF02392726 |mr=0666108| doi-access = free }}.</ref> and thus define special classes of 3- and 4-dimensional submanifolds.
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