| has gloss | eng: In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group G2. The group G_2 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 SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a positive, nondegenerate 3-form, \phi_0. The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in \Lambda^3(\Bbb R^7). The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature (3,4). Thus, G_2 is a subgroup of SO(7). By covariant transport, a manifold with holonomy G_2 has a Riemannian metric and a parallel (covariant constant) 3-form, \phi, the associative form. The Hodge dual, \psi=*\phi 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, respectively. The deformation theory of such submanifolds was studied by McLean. |