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.
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.