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.
A Riemannian or pseudo-Riemannian metric is Einstein when its Ricci tensor is a constant scalar multiple of the metric. Constant-sectional-curvature metrics satisfy this condition with . The Killing-form metric on a real semisimple Lie group is another example, with for in the curvature convention .
The product Riemannian metric of two round unit spheres has Ricci curvature equal to its metric, since the factor Ricci tensors equal their metrics and the product connection splits. Its directional Ricci curvatures are all one. A two-plane tangent to a single sphere has sectional curvature one, while a mixed plane has sectional curvature zero. This four-dimensional Einstein manifold therefore need not have constant sectional curvature.
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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.