Conifold - Revision history

en>Thubsch at 23:44, 9 October 2014

2014-10-09T23:44:33Z

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en>BD2412: Fixing links to disambiguation pages using AWB

2014-02-23T04:09:35Z

Fixing links to disambiguation pages using AWB

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	~~In [[differential geometry]] and [[mathematical physics]], an '''Einstein manifold''' is a [[Riemannian manifold\|Riemannian]] or [[pseudo-Riemannian 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 the dimension, as well as the signature, of the metric can~~ be ~~arbitrary, unlike the four-dimensional [[Lorentzian manifold]]s usually studied in [[general relativity]].~~		There is nothing to say about myself at all.<br>Nice to be here and a member of this site.<br>I really hope I'm useful in one way .<br><br>Also visit my homepage ... [http://designstudiomb.com/content/online-shopping-learn-it-all-right-here coupon 4inkjets discount printer supplies coupon code]

	~~If ''M'' is the underlying ''n''-dimensional [[manifold]]~~ and ~~''g'' is its [[metric tensor]] the Einstein condition means that~~

	~~:<math>\mathrm{Ric} = k\,g,</math>~~

	~~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~~

	:<~~math~~>~~R_{ab} = k\,g_{ab}.</math>~~

	~~Taking the trace of both sides reveals that the constant of proportionality ''k'' for Einstein manifolds is related to the [[scalar curvature]] ''R'' by~~

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

	~~where ''n'' is the dimension of~~ '~~'M''.~~

	~~In [[general relativity]], [[Einstein's equation]] with a [[cosmological constant]] Λ is~~

	~~:<math>R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, </math>~~

	~~written~~ in ~~[[geometrized units]] with ''G'' = ''c'' = 1~~. ~~The [[stress-energy tensor]] ''T''~~<~~sub~~>~~''ab''~~<~~/sub~~> ~~gives the matter and energy content of the underlying spacetime~~. ~~In a [[vacuum]] (a region of spacetime with no matter) ''T''<sub>''ab''</sub> = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2):~~
	~~:<math>R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}~~.~~</math>~~
	~~Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant~~.

	~~== Examples ==~~

	~~Simple examples of Einstein manifolds include:~~

	*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\|''n''-sphere]], ''S''<sup>''n''</~~sup>, 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'''<sup>''n''</~~sup>, with the [[Fubini~~-~~Study metric]].~~
	* [[Calabi Yau manifold]]s admit an Einstein metric that is also [[Kähler metric\|Kähler]], with Einstein constant "k"="0". 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, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.~~

	~~A necessary condition for [[closed manifold\|closed]], [[oriented]], [[4-manifold]]s to be Einstein is satisfying the [[Hitchin–Thorpe inequality]].~~

	~~==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 usually used restricted to Einstein 4-manifolds whose [[Weyl 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 metric\|complete]] but [[compact space\|non-compact]]). In differential geometry, self-~~dual Einstein 4~~-~~manifolds are also known as (4~~-~~dimensional) [[hyperkähler manifold]]s in the Ricci-flat case, and [[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.

	~~==See also==~~

	*[[Einstein–Hermitian vector bundle]]

	~~== 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}}

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

en>FrescoBot: Bot: fixing section wikilinks

2013-12-14T09:59:48Z

Bot: fixing section wikilinks

← Older revision		Revision as of 11:59, 14 December 2013
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			In [[differential geometry]] and [[mathematical physics]], an '''Einstein manifold''' is a [[Riemannian manifold\|Riemannian]] or [[pseudo-Riemannian 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 the dimension, as well as the signature, of the metric can be arbitrary, unlike the four-dimensional [[Lorentzian manifold]]s usually studied in [[general relativity]].

			If ''M'' is the underlying ''n''-dimensional [[manifold]] and ''g'' is its [[metric tensor]] the Einstein condition means that

	~~Title~~ of the ~~author~~ is ~~Gabrielle Lattimer~~. ~~For years she~~'~~s been working compared~~ to a [~~http~~:/~~/dict~~.~~leo.org~~/~~?search~~=~~library+assistant library assistant~~]. ~~For~~ a ~~while she~~'s ~~only been~~ in [~~http~~:/~~/Www~~.~~Thefreedictionary~~.~~com~~/~~Massachusetts Massachusetts~~]. As a ~~woman what your girl really likes~~ is ~~mah jongg but she has~~ not ~~made a dime utilizing~~ it. ~~She has become running~~ and ~~maintaining~~ a ~~meaningful blog here: http://circuspartypanama~~.~~com<br><br>Feel free to visit my website~~ :: [~~http~~:~~//circuspartypanama.com clash of clans hack download android~~]		:<math>\mathrm{Ric} = k\,g,</math>

			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

			:<math>R_{ab} = k\,g_{ab}.</math>

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

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

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

			In [[general relativity]], [[Einstein's equation]] with a [[cosmological constant]] Λ is

			:<math>R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, </math>

			written in [[geometrized units]] with ''G'' = ''c'' = 1. The [[stress-energy tensor]] ''T''<sub>''ab''</sub> gives the matter and energy content of the underlying spacetime. In a [[vacuum]] (a region of spacetime with no matter) ''T''<sub>''ab''</sub> = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2):
			:<math>R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.</math>
			Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant.

			== Examples ==

			Simple examples of Einstein manifolds include:

			*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\|''n''-sphere]], ''S''<sup>''n''</sup>, 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'''<sup>''n''</sup>, with the [[Fubini-Study metric]].
			* [[Calabi Yau manifold]]s admit an Einstein metric that is also [[Kähler metric\|Kähler]], with Einstein constant "k"="0". 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, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.

			A necessary condition for [[closed manifold\|closed]], [[oriented]], [[4-manifold]]s to be Einstein is satisfying the [[Hitchin–Thorpe inequality]].

			==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 usually used restricted to Einstein 4-manifolds whose [[Weyl 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 metric\|complete]] but [[compact space\|non-compact]]). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) [[hyperkähler manifold]]s in the Ricci-flat case, and [[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.

			==See also==

			*[[Einstein–Hermitian vector bundle]]

			== 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}}

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

en>Helpful Pixie Bot: ISBNs (Build KH)

2012-05-11T21:59:45Z

ISBNs (Build KH)

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