On complete hypersurfaces with harmonic curvature in a Riemannian manifold of constant curvature (Q1094687)

Let \(\bar M\) be a space of constant curvature and M a hypersurface with harmonic curvature, i.e., its Ricci tensor is a Codazzi tensor. In a previous paper the second author proved that M has to be an isoparametric hypersurface with at most two principal curvatures, if its mean curvature is constant [see Tsukuba J. Math. 10, 285-292 (1986; Zbl 0616.53042)]. In the paper under review the authors show that this last condition is satisfied if the multiplicity of each principal curvature is at least 2. In this situation the Ricci tensor of M in fact is parallel. But the authors also prove the existence of hypersurfaces with harmonic curvature and non-parallel Ricci tensor.