Harmonic 4-Spaces.
We present short direct proofs of two known properties of complete flat manifolds. They say that the diffeomorphism classes of m-dimensional complete flat manifolds form a finite set and that each element of is represented by a manifold with finite holonomy group.
A Stiefel manifold is the set of orthonormal -frames in , and it is diffeomorphic to the homogeneous space . We study -invariant Einstein metrics on this space. We determine when the standard metric on is Einstein, and we give an explicit solution to the Einstein equation for the space .