Editing Einstein manifold

{{Short description|Riemannian manifold which satisfies vacuum Einstein equations}}
In [[differential geometry]] and [[mathematical physics]], an '''Einstein manifold''' is a [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[differentiable manifold]] whose [[Ricci tensor]] is proportional to the [[Metric tensor|metric]]. 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 [[Lorentzian manifold]]s (including the four-dimensional Lorentzian manifolds usually studied in [[general relativity]]). Einstein manifolds in four Euclidean dimensions are studied as [[gravitational instanton]]s.

If <math>M</math> is the underlying <math>n</math>-dimensional [[manifold]], and <math>g</math> is its [[metric tensor]], the Einstein condition means that

:<math>\mathrm{Ric} = kg</math>

for some constant <math>k</math>, where <math>\operatorname{Ric}</math> denotes the [[Ricci tensor]] of <math>g</math>. Einstein manifolds with <math>k = 0</math> are called [[Ricci-flat manifold]]s.

==The Einstein condition and Einstein's equation==

In local coordinates the condition that <math>(M, g)</math> be an Einstein manifold is simply

:<math>R_{ab} = kg_{ab} .</math>

Taking the trace of both sides reveals that the constant of proportionality <math>k</math> for Einstein manifolds is related to the [[scalar curvature]] <math>R</math> by

:<math>R = nk ,</math>

where <math>n</math> is the dimension of <math>M</math>.

In [[general relativity]], [[Einstein's equation]] with a [[cosmological constant]] <math>\Lambda</math> is

:<math>R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = \kappa T_{ab}, </math>

where <math>\kappa</math> is the [[Einstein gravitational constant]].<ref><math>\kappa</math> should not be confused with <math>k</math>.</ref> The [[stress–energy tensor]] <math>T_{ab}</math> gives the matter and energy content of the underlying spacetime. In [[vacuum]] (a region of spacetime devoid of matter) <math>T_{ab} = 0</math>, and Einstein's equation can be rewritten in the form (assuming that <math>n > 2</math>):
:<math>R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab} .</math>
Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with <math>k</math> proportional to the cosmological constant.

== Examples ==

Simple examples of Einstein manifolds include:
* All 2D manifolds admit Einstein metrics. In fact, in this dimension, a metric is Einstein if and only if it has constant Gauss curvature. The classical uniformization theorem for Riemann surfaces guarantees that there is such a metric in every conformal class on any 2-manifold. 
*Any manifold with [[constant sectional curvature]] is an Einstein manifold&mdash;in particular:
** [[Euclidean space]], which is flat, is a simple example of Ricci-flat, hence Einstein metric.
** The [[n-sphere|''n''-sphere]], <math>S^n</math>, with the round metric is Einstein with <math>k=n-1</math>.
** [[Hyperbolic space]] with the canonical metric is Einstein with <math>k = -(n - 1)</math>.
* [[Complex projective space]], <math>\mathbf{CP}^n</math>, with the [[Fubini–Study metric]], have <math>k = 2n + 2.</math>
* [[Calabi–Yau manifold]]s admit an Einstein metric that is also [[Kähler metric|Kähler]], with Einstein constant <math>k=0</math>. Such metrics are not unique, but rather come in families; there is a Calabi–Yau metric in every Kähler class, and the metric also depends on the choice of complex structure. For example, there is a 60-parameter family of such metrics on [[K3 surface|K3]], 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.
* [[Kähler–Einstein metric]]s exist on a variety of compact [[complex manifold]]s due to the existence results of [[Shing-Tung Yau]], and the later study of [[K-stability]] especially in the case of [[K-stability of Fano varieties|Fano manifolds]]. 
* Irreducible symmetric spaces, as classified by Elie Cartan,  are always Einstein. Among these spaces, the compact ones all have positive Einstein constant <math>k</math>. Examples of these include the Grassmannians <math>Gr (k, \mathbb{R}^\ell)</math>, <math>Gr (k, \mathbb{C}^\ell)</math>, and <math>Gr (k, \mathbb{H}^\ell)</math>. Every such compact space has a so-called non-compact dual, which instead has negative Einstein constant <math>k</math>. These dual pairs are related in manner that is  exactly parallel to the relationship between spheres and hyperbolic spaces.

One necessary condition for [[closed manifold|closed]], [[oriented]], [[4-manifold]]s to be Einstein is satisfying the [[Hitchin–Thorpe inequality]]. However, this necessary condition is very far from sufficient, as further obstructions have been discovered by 
LeBrun, Sambusetti, and others.

==Applications==

Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as [[gravitational instantons]] in [[quantum gravity|quantum theories of gravity]]. The term "gravitational instanton" is sometimes restricted to Einstein 4-manifolds whose [[Weyl tensor]] is anti-self-dual, and it is very often  assumed that the metric is asymptotic to the standard metric on a finite quotient  Euclidean 4-space (and are therefore [[complete metric|complete]] but [[compact space|non-compact]]). In differential geometry, simply connected self-dual Einstein 4-manifolds are coincide with the 4-dimensional, reverse-oriented [[hyperkähler manifold]]s in the Ricci-flat case, but are sometimes called  [[quaternion Kähler manifold]]s otherwise.

Higher-dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as [[string theory]], [[M-theory]] and [[supergravity]]. Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for [[nonlinear σ-model]]s with [[supersymmetry]].

Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging. Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author [[Arthur Besse]], readers are offered a meal in a [[Michelin star|starred restaurant]] in exchange for a new example.<ref>{{harvtxt|Besse|1987|p=18}}</ref>

==See also==

*[[Einstein–Hermitian vector bundle]]
*[[Osserman manifold]]

== Notes and references ==
{{Reflist}}

*{{cite book | first = Arthur L. | last = Besse | authorlink = Arthur Besse | title = Einstein Manifolds | series = Classics in Mathematics | publisher = Springer | location = Berlin | year = 1987 | isbn = 3-540-74120-8}}

[[Category:Riemannian manifolds]]
[[Category:Albert Einstein|Manifold]]
[[Category:Mathematical physics]]