G2 manifold

DISPLAYTITLE:G2 manifold In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in 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 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 Reese Harvey and H. Blaine Lawson, and thus define special classes of 3- and 4-dimensional submanifolds. All -manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers. The fact that might possibly be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this remained consistent with the simplified proof later given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan nonetheless made a useful contribution by showing that, if such a manifold did in fact exist, it would carry both a parallel 3-form and a parallel 4-form, and that it would necessarily be Ricci-flat. The first local examples of 7-manifolds with holonomy were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987. Next, complete (but still noncompact) 7-manifolds with holonomy were constructed by Bryant and Simon Salamon in 1989. The first compact 7-manifolds with holonomy were constructed by Dominic Joyce in 1994. Compact manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature. In 2013, it was shown by M.