In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in 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 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
ϕ
, the associative form. The Hodge dual,
ψ
=
∗
ϕ
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.
If M is a
G
2
-manifold, then M is:
Ricci-flat,
orientable,
a spin manifold.
A manifold with holonomy
G
2
was first introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy
G
2
were constructed by Robert Bryant and Salamon in 1989. The first compact 7-manifolds with holonomy
G
2
were constructed by Dominic Joyce in 1994, and compact
G
2
manifolds are sometimes known as "Joyce manifolds", especially in the physics literature. In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a
G
2
-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with
G
2
-structure. In the same paper, it was shown that certain classes of
G
2
-manifolds admit a contact structure.
These manifolds are important in string theory. They break the original supersymmetry to 1/8 of the original amount. For example, M-theory compactified on a
G
2
manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry. The resulting low energy effective supergravity contains a single supergravity supermultiplet, a number of chiral supermultiplets equal to the third Betti number of the
G
2
manifold and a number of U(1) vector supermultiplets equal to the second Betti number.
