Einstein manifold

Encyclopedia : E : EI : EIN : Einstein manifold

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

[\mathrm = k\,g]

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.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.