-structure is an important type of G-structure that can be defined on a smooth manifold.
If M is a smooth manifold of dimension seven, then a G2-structure is a reduction of structure group of the frame bundle of M to the compact, exceptional Lie group G2.
-structure is much weaker than the existence of a metric of holonomy
must also have finite fundamental group and non-vanishing first Pontrjagin class.
was first suggested by Marcel Berger's 1955 classification of possible Riemannian holonomy groups.
Although thil working in a complete absence of examples, Edmond Bonan then forged ahead in 1966, and investigated the properties that a manifold of holonomy
would necessarily have; in particular, he showed that such a manifold would carry a parallel 3-form and a parallel 4-form, and that the manifold would necessarily be Ricci-flat.
[1] However, it remained unclear whether such metrics actually existed until Robert Bryant proved a local existence theorem for such metrics in 1984.
were constructed by Bryant and Simon Salamon in 1989.
were constructed by Dominic Joyce in 1994, and compact
[3] In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a
-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with
The letter "G" occurring in the phrases "G-structure" and "
In the first case, G-structures take their name from the fact that arbitrary Lie groups are typically denoted with the letter "G".
" comes from the fact that its Lie algebra is the seventh type ("G" being the seventh letter of the alphabet) in the classification of complex simple Lie algebras by Élie Cartan.