In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Spin(7)-manifold is an eight-dimensional Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 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

Dominic Joyce

Dominic D. Joyce 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. He undertook a DPhil in geometry under the supervision of Simon Donaldson, completed in 1992...

in 1996.