G₂ Manifold

G₂ Manifold

In differential geometry, a G₂ manifold is a seven-dimensional Riemannian manifold with holonomy group G₂. 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 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, . The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in . The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature . Thus, is a subgroup of . By covariant transport, a manifold with holonomy has a Riemannian metric and a parallel (covariant constant) 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, respectively. The deformation theory of such submanifolds was studied by McLean.