Compact Manifolds with Exceptional Holonomy

In the theory of Riemannian holonomy groups, the most mysterious are the two exceptional cases, the holonomy group G2 in 7 dimensions and the holonomy group Spin. In a series of three recent papers the author constructed the first known examples of compact 7-manifolds and 8-manifolds with metrics of holonomy G2 and Spin. This chapter gives a brief and informal description of the construction. It also gives Berger’s classification of holonomy groups. For some time after Berger’s classification, the holonomy groups G2 and Spin remained a mystery. In 1987, L. Bryant used the theory of exterior differential systems to show that locally there exist many metrics with these holonomy groups, and gave some explicit, incomplete examples. Then in 1989, Bryant and S. M. Salamon found explicit, complete metrics with holonomy G2 and Spin on non-compact manifolds. The chapter considers the connections with physics.