Einstein manifold

Einstein metricEinsteinEinstein manifoldsEinstein metricsEinstein spaceEinstein spaces
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.wikipedia
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Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.
If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf.

Scalar curvature

Ricci scalarcurvaturecurvature scalar
Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by
The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics.

Ricci-flat manifold

Ricci-flatRicci flatRicci-flat condition
Einstein manifolds with k = 0 are called Ricci-flat manifolds.
Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish.

Einstein field equations

Einstein field equationEinstein's field equationsEinstein's field equation
They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.
The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.

Hitchin–Thorpe inequality

A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.
In differential geometry the Hitchin–Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric.

Gravitational instanton

asymptotically locally EuclideanEguchi-Hansongravitational instantons
Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.
Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds.

Weyl tensor

Weyl curvatureWeyl curvature tensorWeyl
The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensor is self-dual, and it is usually assumed that the metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact).
More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.

Kähler manifold

Kähler metricKählerKähler form
See the article on Einstein manifolds for more details.

Quaternion-Kähler manifold

quaternionic Kähler manifoldquaternion Kähler manifold
In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise.
Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies as the only class of irreducible, non-symmetric manifolds of special holonomy that are automatically Einstein, but not automatically

Fubini–Study metric

Fubini-Study metricFubini–Studyquantum angle
The Fubini–Study metric is also an Einstein metric in that it is proportional to its own Ricci tensor: there exists a constant \Lambda; such that for all i,j we have

Einstein–Hermitian vector bundle

Hermitian–Einstein connectionHermitian–Einstein metric
*Einstein–Hermitian vector bundle
(These examples are however dangerously misleading, because there are compact Einstein manifolds, such as the Page metric on, that are Hermitian, but for which the Levi-Civita connection is not Hermite-Einstein.)

Differential geometry

differentialdifferential geometerdifferential geometry and topology
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

Mathematical physics

mathematical physicistmathematicalmathematical physicists
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

Metric tensor

metricmetricsround metric
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. If M is the underlying n-dimensional manifold and g is its metric tensor the Einstein condition means that

Albert Einstein

EinsteinEinsteinianA. Einstein
They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

Vacuum

free spaceevacuatedhigh vacuum
They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

Cosmological constant

cosmological constant problemcosmological termexpansion
They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

General relativity

general theory of relativitygeneral relativity theoryrelativity
They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

Manifold

manifoldsboundarymanifold with boundary
If M is the underlying n-dimensional manifold and g is its metric tensor the Einstein condition means that

Stress–energy tensor

energy–momentum tensorenergy-momentum tensorstress-energy tensor
The stress–energy tensor T ab gives the matter and energy content of the underlying spacetime.

Closed manifold

closedcompact manifoldcompact
A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.

Orientability

orientableorientednon-orientable
A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.