A positive Ricci curvature problem from Peter Petersen's book

Question

The book "Riemannian Geoemetry, the third edition" by Peter Petersen says the following on page 304: $S^k \times S^1$ does not admit any Ricci flat metrics when $k=2, 3.$ My question is whether $S^k \times S^1$ admits metrics of positive Ricci curvature when $k=2, 3.$ If it is correct, how to prove it?

Answer 1

No. If a manifold $M$ has positive Ricci curvture, then it follows from Myers's theorem that its fundamental group $\pi_1(M)$ is finite. But the fundamental group of $S^k\times S^1$ is $\mathbb{Z}$ for any $k\geq 2$.