932:
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
813:. Such metrics are not unique, but rather come in families; there is a CalabiâYau metric in every KĂ€hler class, and the metric also depends on the choice of complex structure. For example, there is a 60-parameter family of such metrics on
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921:. HyperkÀhler and quaternion KÀhler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for
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is self-dual, and it is usually assumed that the metric is asymptotic to the standard metric of
Euclidean 4-space (and are therefore
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573:
All 2D manifolds are trivially
Einstein manifolds. This is a result of the Riemann tensor having a single degree of freedom.
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76:), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to
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817:, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.
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Higher-dimensional
Lorentzian Einstein manifolds are used in modern theories of gravity, such as
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1007:
898:). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional)
886:. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose
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878:
Four dimensional
Riemannian Einstein manifolds are also important in mathematical physics as
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Therefore, vacuum solutions of
Einstein's equation are (Lorentzian) Einstein manifolds with
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because this condition is equivalent to saying that the metric is a solution of the
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Taking the trace of both sides reveals that the constant of proportionality
586:, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
423:{\displaystyle R_{ab}-{\frac {1}{2}}g_{ab}R+g_{ab}\Lambda =\kappa T_{ab},}
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of a conformal class, rather than the Levi-Civita connection of a metric.
590:
100:
80:(including the four-dimensional Lorentzian manifolds usually studied in
471:, and Einstein's equation can be rewritten in the form (assuming that
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66:
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gives the matter and energy content of the underlying spacetime. In
84:). Einstein manifolds in four Euclidean dimensions are studied as
27:
Riemannian manifold which satisfies vacuum
Einstein equations
547:{\displaystyle R_{ab}={\frac {2\Lambda }{n-2}}\,g_{ab}.}
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846:is a generalization of an Einstein manifold for a
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569:Simple examples of Einstein manifolds include:
181:The Einstein condition and Einstein's equation
1006:. Classics in Mathematics. Berlin: Springer.
580:is an Einstein manifold—in particular:
8:
661:with the canonical metric is Einstein with
561:proportional to the cosmological constant.
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623:, with the round metric is Einstein with
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459:(a region of spacetime devoid of matter)
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259:for Einstein manifolds is related to the
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185:In local coordinates the condition that
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783:admit an Einstein metric that is also
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111:, the Einstein condition means that
937:, readers are offered a meal in a
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197:be an Einstein manifold is simply
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25:
866:to be Einstein is satisfying the
735:{\displaystyle \mathbf {CP} ^{n}}
144:{\displaystyle \mathrm {Ric} =kg}
951:EinsteinâHermitian vector bundle
827:due to the existence results of
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941:in exchange for a new example.
441:Einstein gravitational constant
245:{\displaystyle R_{ab}=kg_{ab}.}
823:exist on a variety of compact
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1:
974:should not be confused with
902:in the Ricci-flat case, and
578:constant sectional curvature
904:quaternion KĂ€hler manifolds
884:quantum theories of gravity
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854:A necessary condition for
835:especially in the case of
166:. Einstein manifolds with
868:HitchinâThorpe inequality
831:, and the later study of
787:, with Einstein constant
880:gravitational instantons
708:Complex projective space
695:{\displaystyle k=-(n-1)}
180:
158:, where Ric denotes the
86:gravitational instantons
70:Einstein field equations
821:KĂ€hlerâEinstein metrics
61:. They are named after
57:is proportional to the
51:differentiable manifold
844:EinsteinâWeyl geometry
807:
772:
771:{\displaystyle k=n+1.}
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900:hyperkÀhler manifolds
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648:{\displaystyle k=n-1}
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616:{\displaystyle S^{n}}
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325:cosmological constant
296:
294:{\displaystyle R=nk,}
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74:cosmological constant
32:differential geometry
1041:Mathematical physics
1031:Riemannian manifolds
957:Notes and references
791:
781:CalabiâYau manifolds
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445:stressâenergy tensor
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308:is the dimension of
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175:Ricci-flat manifolds
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78:Lorentzian manifolds
36:mathematical physics
806:{\displaystyle k=0}
744:FubiniâStudy metric
321:Einstein's equation
1004:Einstein Manifolds
939:starred restaurant
923:nonlinear Ï-models
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576:Any manifold with
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317:general relativity
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154:for some constant
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95:is the underlying
82:general relativity
825:complex manifolds
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48:pseudo-Riemannian
40:Einstein manifold
16:(Redirected from
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1000:Besse, Arthur L.
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584:Euclidean space
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18:Einstein metric
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837:Fano manifolds
829:Shing-Tung Yau
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990:, p. 18)
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927:supersymmetry
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911:string theory
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109:metric tensor
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99:-dimensional
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935:Arthur Besse
931:
919:supergravity
908:
877:
874:Applications
853:
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160:Ricci tensor
155:
153:
104:
96:
92:
90:
55:Ricci tensor
39:
29:
988:Besse (1987
906:otherwise.
896:non-compact
888:Weyl tensor
864:4-manifolds
833:K-stability
742:, with the
173:are called
1025:Categories
327:Λ is
44:Riemannian
684:−
675:−
640:−
519:−
511:Λ
402:κ
396:Λ
351:−
1002:(1987).
945:See also
915:M-theory
892:complete
860:oriented
565:Examples
101:manifold
746:, have
594:-sphere
439:is the
323:with a
107:is its
1010:
856:closed
785:KĂ€hler
476:> 2
457:vacuum
443:. The
433:where
304:where
103:, and
72:(with
67:vacuum
59:metric
53:whose
925:with
42:is a
38:, an
1008:ISBN
917:and
894:but
589:The
34:and
882:in
842:An
478:):
469:= 0
315:In
266:by
171:= 0
162:of
91:If
46:or
30:In
1027::
929:.
913:,
870:.
862:,
858:,
815:K3
766:1.
710:,
596:,
466:ab
452:ab
319:,
312:.
191:,
177:.
88:.
1016:.
978:.
976:k
971:Îș
839:.
801:0
798:=
795:k
763:+
760:n
757:=
754:k
728:n
723:P
720:C
702:.
690:)
687:1
681:n
678:(
672:=
669:k
655:.
643:1
637:n
634:=
631:k
609:n
605:S
592:n
559:k
542:.
537:b
534:a
530:g
522:2
516:n
508:2
502:=
497:b
494:a
490:R
474:n
462:T
448:T
436:Îș
418:,
413:b
410:a
406:T
399:=
391:b
388:a
384:g
380:+
377:R
372:b
369:a
365:g
359:2
356:1
346:b
343:a
339:R
310:M
306:n
289:,
286:k
283:n
280:=
277:R
264:R
257:k
240:.
235:b
232:a
228:g
224:k
221:=
216:b
213:a
209:R
195:)
193:g
189:M
187:(
169:k
164:g
156:k
139:g
136:k
133:=
129:c
126:i
123:R
105:g
97:n
93:M
20:)
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