In differential geometry, a -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.
The condition of M admitting a structure is equivalent to any of the following conditions:
The last condition above correctly suggests that many manifolds admit -structures.
A manifold with holonomy 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 were constructed by Robert Bryant and Salamon in 1989. [2] The first compact 7-manifolds with holonomy were constructed by Dominic Joyce in 1994, and compact 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 -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with -structure. [4] In the same paper, it was shown that certain classes of -manifolds admit a contact structure.
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.
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau who proved the Calabi conjecture.
In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in a vacuum with vanishing cosmological constant.
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of special orthogonal group SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of the general linear group GL(7) which preserves the non-degenerate 3-form , the associative form. The Hodge dual, is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson, and thus define special classes of 3- and 4-dimensional submanifolds.
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 FM (or GL(M)) of M.
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some . Here Sp(n) is the sub-group of consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic matrix, while the group of unit-length quaternions instead acts on quaternionic -space by right scalar multiplication. The Lie group generated by combining these actions is then abstractly isomorphic to .
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, which is a calibrating form for a special class of submanifolds called Cayley cycles.
In differential geometry, a Sasakian manifold is a contact manifold equipped with a special kind of Riemannian metric , called a Sasakian metric.
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions define integrable almost complex structures.
In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-formφ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure , such that the (2,1)-tensor is skew-symmetric. So,
In gravitation theory, a world manifold endowed with some Lorentzian pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over a world manifold.
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 thirty years later, Edmond Bonan nonetheless made a useful contribution by showing in 1966, that such manifolds would carry at least a parallel 4-form, and would necessarily be Ricci-flat, propelling them as candidates for string theory.
In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free -structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem.
Sema Salur is a Turkish-American mathematician, currently serving as a Professor of Mathematics at the University of Rochester. She was awarded the Ruth I. Michler Memorial Prize for 2014–2015, a prize intended to give a recently promoted associate professor a year-long fellowship at Cornell University; and has been the recipient of a National Science Foundation Research Award beginning in 2017. She specialises in the "geometry and topology of the moduli spaces of calibrated submanifolds inside Calabi–Yau, G2 and Spin(7) manifolds", which are important to certain aspects of string theory and M-theory in physics, theories that attempt to unite gravity, electromagnetism, and the strong and weak nuclear forces into one coherent Theory of Everything.
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