Einstein manifold - Definition

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

<math>Ric = k\,g<math>

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.

Example Usage of Einstein