A characterization of Einstein manifolds

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 $λ$ of $r_{g}$, with respect to $g$, satisfy $λ\geq \frac{1}{n}s_{g}$, then $(M,g)$ is an Einstein manifold, where $r_{g}$ and $s_{g}$ denote the Ricci and scalar curvatures, respectively.

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