G₂-structure

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 condition of M admitting a ${\displaystyle G_{2}}$  structure is equivalent to any of the following conditions:

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

The last condition above correctly suggests that many manifolds admit ${\displaystyle G_{2}}$ -structures.

History

A manifold with holonomy ${\displaystyle G_{2}}$  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.^[1] The first complete, but noncompact 7-manifolds with holonomy ${\displaystyle G_{2}}$  were constructed by Robert Bryant and Salamon in 1989.^[2] 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.^[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 ${\displaystyle G_{2}}$ -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with ${\displaystyle G_{2}}$ -structure.^[4] 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, a ${\displaystyle G_{2}}$ -manifold is a manifold with a ${\displaystyle G_{2}}$ -structure which 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., International Press of Boston, 17 (2): 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.