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In differential geometry, a ${\displaystyle G_{2}}$-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.

Equivalent conditions

The existence of a ${\displaystyle G_{2}}$ structure on a 7-manifold ${\displaystyle M}$ is equivalent to either of the following conditions:

• The first and second Stiefel–Whitney classes of M vanish.
• M is orientable and admits a spin structure.

It follows that the existence of a ${\displaystyle G_{2}}$-structure is much weaker than the existence of a metric of holonomy ${\displaystyle G_{2}}$, because a compact 7-manifold of holonomy ${\displaystyle G_{2}}$ must also have finite fundamental group and non-vanishing first Pontrjagin class.

History

The fact that there might be certain Riemannian 7-manifolds manifolds of holonomy ${\displaystyle G_{2}}$ 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 ${\displaystyle G_{2}}$ 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. However, it remained unclear whether such metrics actually existed until Robert Bryant proved a local existence theorem for such metrics in 1984. The first complete (although non-compact) 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Bryant and Simon Salamon in 1989. The first compact 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Dominic Joyce in 1994, and compact ${\displaystyle G_{2}}$ 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 ${\displaystyle G_{2}}$-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with ${\displaystyle G_{2}}$-structure. In the same paper, it was shown that certain classes of ${\displaystyle G_{2}}$-manifolds admit a contact structure.

Remarks

The property of being a ${\displaystyle G_{2}}$-manifold is much stronger than that of admitting a ${\displaystyle G_{2}}$-structure. Indeed, being a ${\displaystyle G_{2}}$-manifold is equivalent to admitting a ${\displaystyle G_{2}}$-structure that is torsion-free.

The letter "G" occurring in the phrases "G-structure" and "${\displaystyle G_{2}}$-structure" refers to different things. In the first case, G-structures take their name from the fact that arbitrary Lie groups are typically denoted with the letter "G". On the other hand, the letter "G" in "${\displaystyle G_{2}}$" 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.

See also

• G₂, G₂-manifold, Spin(7) manifold

Notes

1. E. Bonan (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris, 262: 127–129.
2. Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58 (3): 829–850, doi:10.1215/s0012-7094-89-05839-0.
3. Joyce, D.D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
4. Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on ${\displaystyle G_{2}}$-manifolds", Asian J. Math., 17 (2), International Press of Boston: 321–334, arXiv:1112.2951, doi:10.4310/AJM.2013.v17.n2.a3, S2CID 54942812.

References

• Bryant, R. L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.

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