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.

History

A 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.
• M.Fernandez (1986), "A classification of Riemannian with structure Spin(7)", Annali di Matematica pura ed applicata, doi:10.1007/bf01769211.
• Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal 58: 829–850, doi:10.1215/s0012-7094-89-05839-0.
• Dominic Joyce (2000). Compact Manifolds with Special Holonomy. Oxford University Press. ISBN 0-19-850601-5.
• Karigiannis, Spiro, "Flows of G2 and Spin(7) structures" (PDF), Mathematical Institute, University of Oxford.