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
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.
-manifold are 7-dimensional, Ricci-flat, orientable spin manifolds.
In addition, any compact manifold with holonomy equal to
has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers.
might possibly be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this remained consistent with the simplified proof later given by Jim Simons in 1962.
Although not a single example of such a manifold had yet been discovered, Edmond Bonan nonetheless made a useful contribution by showing that, if such a manifold did in fact exist, it would carry both a parallel 3-form and a parallel 4-form, and that it would necessarily be Ricci-flat.
were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987.
[5] 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
manifolds, due to Alessio Corti, Mark Haskins, Johannes Nordstrőm, and Tommaso Pacini, combined a gluing idea suggested by Simon Donaldson with new algebro-geometric and analytic techniques for constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples.
manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry.
The resulting low energy effective supergravity contains a single supergravity supermultiplet, a number of chiral supermultiplets equal to the third Betti number of the
Recently it was shown that almost contact structures (constructed by Sema Salur et al.)[6] play an important role in