nLab
Einstein manifold

Context

Riemannian geometry

Gravity

Constant

Definition

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

Ric=λgSym 2Γ(TX) Ric = \lambda g \in Sym^2 \Gamma(T X)

by a proportionality constant λ\lambda \in \mathbb{R}.

Properties

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

Examples

References