On rotationally invariant shrinking Ricci solitons

We study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimensions n ≥ 3 and prove that the only complete metrics on S n , ‫ޒ‬ n , and ‫ޒ‬ × S n-1 that admit shrinking soliton structures are respectively the round, flat, and standard cylindrical metrics.