  We show that all the small covers which are infra-nilmanifolds are exactly
real Bott manifolds. This implies that any small cover which admits a flat
Riemannian metric must be a real Bott manifold. In addition, we will study
small covers which admit Riemannian metrics with positive or nonnegative Ricci
curvature or sectional curvature. We will see that these geometric conditions
put very strong restrictions on the topology of the small covers and the
combinatorial structure of the underlying simple polytopes. Similar geometric
problems are also studied for the real moment-angle complex of an arbitrary
simple polytope.
