nLab Einstein manifold

Constant

Formalism

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

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.

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

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 are Einstein manifolds see there

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 August 6, 2023 at 01:37:33. See the history of this page for a list of all contributions to it.