# nLab Einstein manifold

Constant

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Gravity

gravity, supergravity

# Constant

## 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 $\lambda \in \mathbb{R}$.

## 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 $\lambda$.

• all quaternion-Kähler manifolds are Einstein manifolds see there

## 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)

Last revised on September 2, 2021 at 06:34:51. See the history of this page for a list of all contributions to it.