nLab Einstein manifold

Definition

An Einstein manifold is a (pseudo-)Riemannian manifold $(X,g)$ (a spacetime) such that the Ricci tensor is proportional to the metric tensor

$$Ric = \lambda g \in Sym^2 \Gamma(T X)$$

by a proportionality constant $\lambda \in \mathbb{R}$. Such a metric $g$ is called an Einstein metric.

Properties

Einstein manifolds are precisely the solutions of Einstein's equations for pure gravity with cosmological constant $\lambda$.

Examples

• A manifold of dimension 7 and of weak G₂-holonomy with weakness parameter $\lambda$ – $d \omega_3 = \lambda \star \omega_3$ – is canonically an Einstein manifold with cosmological constant $\lambda$.

• de Sitter spacetime, anti de Sitter spacetime

• all quaternion-Kähler manifolds, see there

Related concepts

• Kähler-Einstein manifold

References

• Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Diff Geom. 9 (1974) 435-441 [pdf]

• Jongmin Park, Jaewon Shin, Hyun Seok Yang, Unification of Einstein Manifolds [arXiv:2109.00001]