Codazzi Tensors with Two Eigenvalue Functions
This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is n -1, then the metric is a warped product where the base is an open interval. Tojeiro generalized this result given the existence of a certain type of function of the eigenvalues of the Codazzi tensor, of which the trace is one example. We will show the equivalence of Tojeiro's condition to three other conditions. Furthermore, we construct examples of Codazzi tensors
having two eigenvalue functions, one of which has eigenspace dimension n-1 where the metric is not a warped product with interval base, refuting a remark in Besse that the warped product conclusion holds without any restriction on the trace.