Codazzi Tensors with Two Eigenvalue Functions

TL;DR

This paper investigates the structure of Codazzi tensors with two eigenfunctions on higher-dimensional Riemannian manifolds, relaxing previous trace conditions and providing counterexamples to earlier assumptions about warped product metrics.

Contribution

It extends the classification of Codazzi tensors by weakening trace conditions and constructs counterexamples to prior claims about warped product structures.

Findings

Under milder trace conditions, the metric is a warped product with an interval base.

Counterexamples show that the warped product conclusion does not hold without restrictions on the trace.

The paper refutes a previous remark suggesting the warped product structure always holds for such tensors.

Abstract

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- 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 $n-1$, where the metric is not a warped product with interval base, refuting a remark in \cite{Besse} that the warped product conclusion holds without any restriction on the trace.