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G2 manifold
In differential geometry, a ''G''2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in ''G''2. The group is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, 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 , the associative form. The Hodge dual, is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Harvey–Lawson, and thus define special classes of 3- and 4-dimensional submanifolds.
== Properties ==
If ''M'' is a -manifold, then ''M'' is:
* Ricci-flat,
* orientable,
* a spin manifold.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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