G2-structure

In differential geometry, a G₂-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 G₂ 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 G₂-structures.

History

Manifolds with holonomy $G_2$ were first introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that these manifolds are 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.

Remarks

The property of being a G₂-manifold is much stronger than that of admitting a G₂-structure. Indeed, a G₂-manifold is a manifold with a G₂-structure which is torsion-free.

The letter "G" occurring in the phrases "G-structure" and "G₂-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 "G₂" comes from the fact that the 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-manifold, Spin(7) manifold

References

• E. Bonan, (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris 262: 127–129.
• Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics (Annals of Mathematics) 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.
• Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal 58: 829–850, doi:10.1215/s0012-7094-89-05839-0.
• Joyce, D.D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.