Does a compact manifold require non zero Ricci curvature?

Question

Imagine we have a Riemanian compact manifold. Does the compactness necesarily make its curvature non zero? If the answer were no, does anyone know any such manifold with isometry group $U(1)\times{}SU(2)\times{}SU(3)$?

Answer 1

No. In fact compact Riemannian manifolds can have zero sectional curvature; take flat tori $\mathbb{R}^n/\Gamma$ where $\mathbb{R}^n$ has the usual Euclidean metric and $\Gamma$ is a lattice.

Apparently every compact Lie group is the isometry group of some compact Riemannian manifold, but I don't know how cavalier one can be about specifying its curvature.