differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, a

G_2

-structure is an important type of

G-structure In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes vario ...

that can be defined on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

. If ''M'' is a smooth manifold of dimension seven, then a G₂-structure is a reduction of structure group of the

frame bundle In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...

of ''M'' to the

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, exceptional

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

G₂.

Equivalent conditions

The existence of a

G_2

structure on a 7-manifold

M

is equivalent to either of the following conditions: *The first and second

Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...

es of ''M'' vanish. *''M'' is

orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ...

and admits a

spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathemati ...

. It follows that the existence of a

G_2

-structure is much weaker than the existence of a metric of holonomy

G_2

, because a compact 7-manifold of holonomy

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

G_2

was first suggested by

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Biography After studying from 1948 to 19 ...

's 1955 classification of possible Riemannian holonomy groups. Although still working in a complete absence of examples,

Edmond Bonan Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy. Although not a single example of G2 manifold or Spin(7) manifold had been discovered until thir ...

then forged ahead in 1966, and investigated the properties that a manifold of holonomy

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

G_2

were constructed by Bryant and Simon Salamon in 1989. The first compact 7-manifolds with holonomy

G_2

were constructed by

Dominic Joyce Dominic David Joyce FRS (born 8 April 1968) is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College, Oxford. He und ...

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

, 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 In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...

Remarks

The property of being a

G_2

-manifold is much stronger than that of admitting a

G_2

-structure. Indeed, being a

G_2

-manifold is equivalent to admitting a

G_2

-structure that is torsion-free. The letter "G" occurring in the phrases "G-structure" and "

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 "

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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

Notes

References

*. Differential geometry Riemannian geometry Structures on manifolds {{differential-geometry-stub

Equivalent conditions

History

Remarks

See also

Notes

References