In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Spin(7)-manifold is an eight-dimensional
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
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.
History
The fact that Spin(7) might possibly arise as the holonomy group of certain Riemannian 8-manifolds was first suggested by the 1955
classification theorem of
Marcel Berger, and this possibility remained consistent with the simplified proof of Berger's theorem given by
Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered,
Edmond Bonan then showed in 1966 that,
if such a manifold did in fact exist, it would carry a parallel 4-form, and that it would necessarily be Ricci-flat. The first local examples of 8-manifolds with holonomy Spin(7) were finally constructed around 1984 by
Robert Bryant, and his full proof of their existence appeared in Annals of Mathematics in 1987.
[Bryant, Robert L. (1987) "Metrics with exceptional holonomy," ]Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
(2)126, 525–576. Next, complete (but still noncompact) 8-manifolds with holonomy Spin(7) were explicitly constructed by Bryant and Salamon in 1989. The first examples of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Spin(7)-manifolds were then constructed by
Dominic Joyce in 1996.
See also
*
''G''2 manifold
*
Calabi–Yau manifold
References
*.
*.
*
*.
Riemannian manifolds
{{Riemannian-geometry-stub