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,[1] and thus define special classes of 3- and 4-dimensional submanifolds.
differential geometry, a G2manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G 2 {\displaystyle...
groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. See G2manifold and Spin(7) manifold. Note that Sp(n) ⊂...
the form of a Calabi–Yau manifold. Within the more complete framework of M-theory, they would have to take form of a G2manifold. A particular exact symmetry...
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...
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...
G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds were...
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...
of the group of rotations of the E8 lattice. G2manifold Octonion algebra Okubo algebra Spin(7) manifold Spin(8) Split-octonions Triality Sabadini, Irene;...
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...
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...
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