A note on Codazzi tensors

We discuss a gap in Besse’s book (Einstein manifolds, 2008), recently pointed out by Merton in (Proc Am Math Soc 141:3265–3273, 2013), which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons.