In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a
-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 var ...
that can be defined on a
smooth manifold. If ''M'' is a smooth manifold of dimension seven, then a G
2-structure is a reduction of structure group of the
frame bundle 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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, exceptional
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
G2.
Equivalent conditions
The condition of ''M '' admitting a
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
-structures.
History
A manifold with holonomy
was first introduced by
Edmond Bonan
Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy.
Biography
After completing his undergraduate studies ...
in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy
were constructed by
Robert Bryant and Salamon in 1989. The first compact 7-manifolds with holonomy
were constructed by
Dominic Joyce
Dominic David Joyce Fellow of the Royal Society, FRS (born 8 April 1968) is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College, Oxford, Lincoln College since 1995. His undergraduate and doc ...
in 1994, and compact
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
-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with
-structure.
[.] In the same paper, it was shown that certain classes of
-manifolds admit a
contact structure.
Remarks
The property of being a
-manifold is much stronger than that of admitting a
-structure. Indeed, a
-manifold is a manifold with a
-structure which is
torsion-free.
The letter "G" occurring in the phrases "G-structure" and "
-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 "
" 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. ...
.
See also
*
G2,
G2-manifold,
Spin(7) manifold
In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, ...
Notes
References
*.
Differential geometry
Riemannian geometry
Structures on manifolds
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