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.
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.