An Einstein manifold is a Riemannian manifold where the Ricci curvature is proportional to the metric tensor. This means that for an Einstein manifold, the Ricci tensor can be expressed as $R_{ij} = \lambda g_{ij}$, where $\lambda$ is a constant and $g_{ij}$ is the metric tensor. The concept of Einstein manifolds is crucial in understanding the relationship between geometry and the distribution of matter in general relativity.
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