Spin(7)-manifold

In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold with the exceptional holonomy group 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. The deformation theory of such submanifolds was first investigated by R. McLean.

History

Manifold with holonomy Spin(7) was firstly introduced by Edmond Bonan in 1966, who constructed the parallel 4-form and showed that this manifold was Ricci-flat. Examples of complete Spin(7)-metrics on non-compact manifolds were first constructed by Bryant and Salamon in 1989. The first examples of compact Spin(7)-manifolds were constructed by Dominic Joyce in 1996.

See also

• G₂ manifold
• Calabi–Yau manifold

References

• E. Bonan, (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris 262: 127–129 .
• Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal 58: 829–850 .
• Dominic Joyce (2000). Compact Manifolds with Special Holonomy. Oxford University Press. ISBN 0-19-850601-5.