Codazzi tensors with two eigenvalue functions

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Abstract
This paper addresses a gap in the classification 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 eigenspaces is n − 1 n-1 , then the metric is a warped product where the base is an open interval, a conclusion we will show to be true under a milder trace condition. Furthermore, we construct examples of Codazzi tensors having two eigenvalue functions, one of which has eigenspace dimension n − 1 n-1 , where the metric is not a warped product with interval base, refuting a claim by A. L. Besse that the warped product conclusion holds without any restriction on the trace.