  The classification of Riemannian manifolds by the holonomy group of their
Levi-Civita connection picks out many interesting classes of structures,
several of which are solutions to the Einstein equations. The classification
has two parts. The first consists of isolated examples: the Riemannian
symmetric spaces. The second consists of geometries that can occur in
continuous families: these include the Calabi-Yau structures and Joyce
manifolds of string theory. One may ask how one can weaken the definitions and
still obtain similar classifications. We present two closely related
suggestions. The classifications for these give isolated examples that are
isotropy irreducible spaces, and known families that are the nearly K\"ahler
manifolds in dimension 6 and Gray's weak holonomy G$_2$ structures in dimension
7.
