differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G 2 {\displaystyle...

of a G2 manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that G2 manifolds are not complex manifolds, so...

geometry of a Calabi–Yau manifold to noncommutative algebraic geometry. Quintic threefold G2 manifold The Calabi-Yau manifold was the subject of a paper...

Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental...

\mu (G)=5} . G2 is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G2 holonomy are also...

Calabi–Yau manifold Complex manifold, a manifold over the complex numbers Differentiable manifold Einstein manifold Flag manifold Flat manifold G2 manifold Hermitian...

4-manifold. Brieskorn manifold Exotic sphere Homology sphere Homotopy sphere Lens space Spherical 3-manifold Einstein manifold Ricci-flat manifold G2 manifold...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

Dominic Joyce in 1996. G2 manifold Calabi–Yau manifold Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", Comptes...

groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. See G2 manifold and Spin(7) manifold. Note that Sp(n) ⊂...

In differential geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex...

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...

between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

the form of a Calabi–Yau manifold. Within the more complete framework of M-theory, they would have to take form of a G2 manifold. A particular exact symmetry...

describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold  T. The non-linear σ-model was introduced by Gell-Mann &...

Forms and Geometric Invariants," from which the theory arose. Given a manifold and a Lie algebra valued 1-form A {\displaystyle \mathbf {A} } over it...

sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of a completely different Calabi–Yau manifold. This equivalence provides...

G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds were...

dimensions spacetime is folded up to form a small torus or other compact manifold. This would leave only the familiar four dimensions of spacetime visible...

compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

compact extra dimensions must be shaped like a Calabi–Yau manifold. A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

curve). For example, a subclass of the K3 manifolds is elliptically fibered, and F-theory on a K3 manifold is dual to heterotic string theory on a two-torus...

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind...

geometry, quaternion-Kähler geometry, G2 geometry, and Spin(7) geometry, each of which study Riemannian manifolds equipped with certain extra structures...

of the group of rotations of the E8 lattice. G2 manifold Octonion algebra Okubo algebra Spin(7) manifold Spin(8) Split-octonions Triality (Baez 2002,...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7) manifold Generalized complex manifold Orbifold...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...