Ricci solitons on locally conformally flat hypersurfaces in space forms

We study Ricci solitons on locally conformally flat hypersurfaces MnMn in space forms M˜n+1(c) of constant sectional curvature cc with potential vector field a principal curvature eigenvector of multiplicity one. We show that in Euclidean space, MnMn is a hypersurface of revolution given in terms of a solution of some non-linear ODE. Hence there exists infinitely many mutually non-congruent Ricci solitons of this type. Furthermore when c≥0c≥0 and MnMn is complete, the Ricci soliton is gradient and in the case it is shrinking, MnMn must be the product of the real line and the (n−1)(n−1)-sphere.