A characterization of Einstein manifolds

TL;DR

This paper characterizes Einstein manifolds without assuming compactness or unit volume, showing that a curvature eigenvalue condition implies the manifold is Einstein.

Contribution

It provides a new characterization of Einstein manifolds based on eigenvalue conditions of the Ricci tensor without compactness assumptions.

Findings

Eigenvalue condition implies Einstein manifold

No compactness or volume assumptions needed

Generalizes previous characterizations

Abstract

In this work we wish characterize the Einstein manifolds $(M,g)$, however without the necessity of hypothesis of compactness over $M$ and unitary volume of $g$, which are well known in many works. Our result says that if all eingenvalues $\lambda$ of $r_{g}$, with respect to $g$, satisfy $\lambda \geq \frac{1}{n}s_{g}$, then $(M,g)$ is an Einstein manifold, where $r_{g}$ and $s_{g}$ denote the Ricci and scalar curvatures, respectively.