An Einstein manifold has Ricci curvature equal to a constant multiple of its metric. In the Riemannian setting, the round unit sphere has . This condition is weaker than constant sectional curvature in higher dimensions. The definition can also be applied to pseudo-Riemannian metrics, but positivity is assumed in the completeness theorems discussed here.
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An Einstein manifold is a Riemannian manifold \((M, g)\) where the Ricci curvature is proportional to the metric tensor \(g\). Mathematically, this relationship can be expressed as: \[ \text{Ric}(g) = \lambda g \] where \(\text{Ric}(g)\) is the Ricci curvature tensor and \(\lambda\) is a constant, often referred to as the "Einstein constant.