Einstein manifold
From Wikipedia, the free encyclopedia
Consider an m-dimensional manifold M with metric tensor g. The pair (M,g) is said to be an Einstein manifold if the Ricci tensor on M is proportional to the metric:
for some constant k. If one introduces a coordinate chart, the condition that (M,g) be an Einstein manifold is then simply
By tensor multiplying both sides of this equation by gab one shows that the constant of proportionality, k, for Einstein manifolds is related to the scalar curvature by
where R is the scalar curvature. Einstein manifolds with k = 0 are also often referred to as Ricci-flat manifolds.
In general relativity, Einstein's equations relate the curvature of a space-time to the energy-momentum tensor which describes the matter distribution in the space-time according to
where we have used geometrized units G = c = 1. Therefore, in a spacetime one for which Tab = 0 we can regard an Einstein manifold as a vacuum solution to Einstein's equations with a cosmological constant proportional to k.
[edit] 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.
[edit] References
- Arthur L. Besse, "Einstein Manifolds", Springer-Verlag.
 |
This geometry-related article is a stub. You can help Wikipedia by expanding it. |
