On the principal Ricci curvatures of a Riemannian 3-manifold

TL;DR

This paper investigates topological and geometric obstructions to the eigenvalues of the Ricci tensor on 3-manifolds, revealing conditions under which certain Ricci eigenvalue configurations cannot exist or imply specific geometric structures.

Contribution

It provides new topological restrictions on Ricci eigenvalues and characterizes geometric structures when eigenvalues have particular forms, especially involving zero eigenvalues.

Findings

No Riemannian metric with Ricci eigenvalues (-μ,f,f) on closed 3-manifolds.

Complete manifolds with Ricci eigenvalues (0,λ,λ) must have a universal cover splitting isometrically.

Flatness of scalar-flat manifolds with a divergence-free, geodesic zero eigenspace vector field.

Abstract

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form $(-\mu,f,f)$, where $\mu$ is a positive constant and $f$ is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form $(0,\lambda,\lambda)$, where $\lambda$ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.