A note on Codazzi tensors

TL;DR

This paper addresses a classification gap in Riemannian geometry related to Codazzi tensors with two eigenvalues, providing a structure theorem and applications to 3D gradient Ricci solitons.

Contribution

It proves a structure theorem for manifolds with specific Codazzi tensors, filling a gap in existing classification theory without extra assumptions.

Findings

Established a structure theorem for manifolds with two-eigenvalue Codazzi tensors.

Applied the theorem to classify three-dimensional gradient Ricci solitons.

Clarified a previously identified gap in the literature.

Abstract

We discuss a gap in Besse's book, recently pointed out by Merton, 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.