3-Manifolds with Constant Ricci Eigenvalues $(\lambda, \lambda, 0)$

TL;DR

This paper classifies complete 3-manifolds with Ricci eigenvalues $(\lambda,\lambda,0)$, revealing their topological structure and providing descriptions under certain geometric conditions.

Contribution

It offers a classification of such manifolds' topology and describes their metrics when locally irreducible or analytic, extending understanding of Ricci eigenvalue configurations.

Findings

Manifolds have free fundamental groups if non-trivial.

Every free group can be realized as the fundamental group.

Descriptions provided for metrics under specific geometric conditions.

Abstract

We consider complete Riemannian $3$-manifolds whose Ricci tensors have constant eigenvalues $(\lambda, \lambda, 0)$. When $\pi_1$ is finitely generated, we classify the topology of such manifolds by showing that they have a free fundamental group if non-trivial and that every free group is obtained. We give a description up to isometry, when the metric is locally irreducible or when it is analytic.