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An Einstein manifold is a Riemannian manifold whose Ricci tensor is proportional to the metric tensor:

In general relativity, these manifolds (in the pseudo-Riemannian case) can be thought of as vacuum solutions of Einstein's equations with a cosmological constant proportional to k.

Einstein manifolds with k = 0 are also called Ricci-flat manifolds.

Examples

The n- sphere, Sⁿ, with the round metric is Einstein with k = n − 1.
Hyperbolic space with the canonical metric is Einstein with negative k.
Complex projective space, CPⁿ, with the Fubini-Study metric .

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