- Einstein manifold
In

differential geometry andmathematical physics , an**Einstein manifold**is a Riemannian orpseudo-Riemannian manifold whoseRicci tensor is proportional to the metric. They are named afterAlbert Einstein because this condition is equivalent to saying that the metric is a solution of thevacuum Einstein equations (withcosmological constant ), although the dimension, as well as the signature, of the metric can be arbitrary, unlike the four dimensionalLorentzian manifold s usually studied ingeneral relativity .If "M" is the underlying "n"-dimensional

manifold and "g" is itsmetric tensor the Einstein condition means that:$mathrm\{Ric\}\; =\; k,g,$

for some constant "k", where Ric denotes the Ricci tensor of "g". Einstein manifolds with "k" = 0 are called

Ricci-flat manifold s.**The Einstein condition and Einstein's equation**In local coordinates the condition that ("M", "g") be an Einstein manifold is simply

:$R\_\{ab\}\; =\; k,g\_\{ab\}.$

Take the trace of both sides one finds that the constant of proportionality "k" for Einstein manifolds is related to the

scalar curvature "R" by:$R\; =\; nk,$

where "n" is the dimension of "M".

In

general relativity ,Einstein's equation with acosmological constant Λ is:$R\_\{ab\}\; -\; frac\{1\}\{2\}g\_\{ab\}R\; +\; g\_\{ab\}Lambda\; =\; 8pi\; T\_\{ab\},$

written in

geometrized units with "G" = "c" = 1. Thestress-energy tensor "T"_{"ab"}gives the matter and energy content of the underlying spacetime. In avacuum (a region of spacetime with no matter) "T"_{"ab"}= 0, and one can rewrite the Einstein's equation in the form (assuming "n" > 2)::$R\_\{ab\}\; =\; frac\{2Lambda\}\{n-2\},g\_\{ab\}.$Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with "k" proportional to the cosmological constant.**Examples**Some of the simplest examples of Einstein manifolds are the following.

*Any manifold with

constant sectional curvature is an Einstein manifold. In particular:

**Euclidean space , which is flat, is a simple example of Ricci-flat, hence Einstein metric.

** The "n"-sphere, "S"^{"n"}, with the round metric is Einstein with "k" = "n" − 1.

**Hyperbolic space with the canonical metric is Einstein with negative "k".

*Complex projective space ,**CP**^{"n"}, with theFubini-Study metric .

*Calabi Yau manifolds admit a unique Einstein metric which is also Kähler**Applications**Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as

gravitational instantons in quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whoseWeyl tensor is self-dual, and it is usually assumed that metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional)hyperkähler manifold s in the Ricci-flat case, andquaternion Kähler manifold s otherwise.Higher dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as

string theory ,M-theory andsupergravity . Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces fornonlinear σ-model s withsupersymmetry .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 starred restaurant in exchange for a new example.**References***cite book | first = Arthur L. | last = Besse | title = Einstein Manifolds | series = Classics in Mathematics | publisher = Springer | location = Berlin | year = 1987 | isbn = 3-540-74120-8

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