Abstract:

In 1955, Marcel Berger classified all the possible holonomy groups for simply connected, irreducible and nonsymmetric Riemannian manifolds. The exceptional holonomy group Spin(7) appeared in this list. However, no examples of manifolds with such holonomy were known until Bryant provided incomplete examples in 1987, and Bryant–Salamon provided complete ones in 1989.

On each Bryant–Salamon manifold, we describe new and old examples of Cayley submanifolds, which are volume-minimizing submanifolds induced by the holonomy reduction. If time permits, we will also study the geometry of the fibrations that they form.