Einstein manifold

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An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor:

<math>\mathrm{Ric} = k\,g</math>

Metric of such manifolds are called Einstein metric. Taking a trace shows that k is equal to s/n, where n is the dimension of M and s is the scalar curvature. Einstein manifolds with k = 0 are also called Ricci-flat manifolds.

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.

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.

References

• Arthur L. Besse, "Einstein Manifolds", Springer-Verlag.Template:Geometry-stub

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