  We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the
space of metrics, up to scaling and diffeomorphism. We also prove the same
rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed
Einstein manifolds. These results are the Ricci flow analogues of our results
for Clifford-type shrinking solitons for the mean curvature flow.
