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
In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...
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
In mathematics, a classification theorem answers the classification problem: "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few issues rela ...
of
Marcel Berger
Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France.
Biography
After studying from 1948 to 19 ...
, and this possibility remained consistent with the simplified proof of Berger's theorem given by
Jim Simons
James Harris Simons (April 25, 1938 – May 10, 2024) was an American hedge fund manager, investor, mathematician, and philanthropist. At the time of his death, Simons's net worth was estimated to be $31.4 billion, making him the 55th-richest ...
in 1962. Although not a single example of such a manifold had yet been discovered,
Edmond Bonan
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 thir ...
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
Dominic David Joyce FRS (born 8 April 1968) is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College, Oxford. He und ...
in 1996.
See also
*
''G''2 manifold
*
Calabi–Yau manifold
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...
References
*.
*.
*
*.
Riemannian manifolds
{{Riemannian-geometry-stub