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Codazzi tensors with two eigenvalue functions

Author: Gabe Merton
Journal: Proc. Amer. Math. Soc. 141 (2013), 3265-3273
MSC (2010): Primary 53A45, 53B20
Published electronically: May 16, 2013
MathSciNet review: 3068979
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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$, 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 claim by A. L. Besse that the warped product conclusion holds without any restriction on the trace.

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Gabe Merton
Affiliation: Department of Mathematics, University of Calfornia, Los Angeles, Box 951555, Los Angeles, California 90095-1555

Received by editor(s): November 29, 2011
Published electronically: May 16, 2013
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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