Einstein manifold

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.

Einstein manifold Ricci flow Gregorio Ricci-Curbastro Symmetric tensor Comparison theorem

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.

Einstein–Hilbert action Positive energy theorem Einstein field equations Shing-Tung Yau Lagrangian (field theory)

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 manifold Calabi–Yau manifold Hyperkähler manifold Ricci curvature Holonomy

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.

Einstein tensor Stress–energy tensor Spacetime symmetries Rotating black hole Gravitational constant

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.

Einstein manifold Nigel Hitchin Claude LeBrun Differential geometry 4-manifold

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.

Einstein manifold Hyperkähler manifold Fubini–Study metric Eguchi–Hanson space Orbifold

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.

Ricci-flat manifold Hermann Weyl Ricci decomposition Conformally flat manifold Einstein manifold

Kähler manifold

Kähler metricKählerKähler form

See the article on Einstein manifolds for more details.

Hodge theory Differential geometry Plurisubharmonic function Symplectic manifold Hermitian manifold

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

Holonomy Quaternion-Kähler symmetric space Edmond Bonan Einstein manifold Hyperkähler manifold

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

Guido Fubini Metric (mathematics)Complex projective space Projective Hilbert space Eduard Study

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.)

Kobayashi–Hitchin correspondence Einstein manifold Stable vector bundle Holomorphic vector bundle Kähler manifold

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.

Gaspard Monge Differential topology Linear algebra Differentiable manifold Differential calculus

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.

Analytical mechanics Potential theory Homological algebra Category theory Journal of Mathematical Physics

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.

Manifold Differential geometry Metric tensor Riemannian geometry Differential geometry of surfaces

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.

Differential geometry Spacetime Metric tensor General relativity Differentiable manifold

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.

Topological manifold Differential structure Tangent space Differential geometry Vector field

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

Metric (mathematics)Differential geometry Tensor field Line element Tensor

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.

Theory of relativity Mass–energy equivalence General relativity Annus Mirabilis papers Photoelectric effect

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.

Vacuum tube Suction Incandescent light bulb Ground state Vacuum pump

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.

Physical cosmology Vacuum energy Dark energy Lambda-CDM model 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.