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{{DISPLAYTITLE:G<sub>2</sub> manifold}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[differential geometry]], a '''''G''<sub>2</sub> manifold''' is a seven-dimensional [[Riemannian manifold]] with [[holonomy group]] [[G2 (mathematics)|''G''<sub>2</sub>]]. The [[group (mathematics)|group]] <math>G_2</math> is one of the five exceptional [[simple Lie group]]s. It can be described as the [[automorphism group]] of the [[octonions]], or equivalently, as a proper subgroup of SO(7) that preserves a [[spinor]] in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a ''positive, nondegenerate'' 3-form, <math>\phi_0</math>. The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in <math>\Lambda^3(\Bbb R^7)</math>. The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature <math>(3,4)</math>. Thus, <math>G_2</math> is a subgroup of <math>SO(7)</math>. By covariant transport, a manifold with holonomy <math>G_2</math> has a Riemannian metric and a parallel (covariant constant) 3-form, <math>\phi</math>, the associative form. The Hodge dual, <math>\psi=*\phi</math> is then a parallel 4-form, the coassociative form. These forms are [[calibrated geometry|calibrations in the sense of Harvey-Lawson]], and thus define special classes of 3 and 4 dimensional submanifolds, respectively. The deformation theory of such submanifolds was studied by McLean.  


== Properties ==
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If ''M'' is a <math>G_2</math>-manifold, then ''M'' is:
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* [[Ricci-flat]]
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* [[orientable]]
* a [[spin manifold]]


== History ==
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Manifold with holonomy <math>G_2</math> was firstly introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy <math>G_2</math> were constructed by [[Robert Bryant (mathematician)|Robert Bryant]] and Salamon in 1989.  The first compact 7-manifolds with holonomy <math>G_2</math> were constructed by [[Dominic Joyce]] in 1994, and compact <math>G_2</math> manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.


== Connections to physics ==
'''MathML'''
These manifolds are important in [[string theory]]. They break the original [[supersymmetry]] to 1/8 of the original amount. For example, [[M-theory]] compactified on a <math>G_2</math> manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry.  The resulting low energy effective [[supergravity]] contains a single supergravity [[supermultiplet]], a number of [[chiral supermultiplet]]s equal to the third [[Betti number]] of the <math>G_2</math> manifold and a number of U(1) [[vector supermultiplet]]s equal to the second Betti number.
:<math forcemathmode="mathml">E=mc^2</math>


''See also'': [[Calabi-Yau manifold]], [[Spin(7) manifold]]
<!--'''PNG'''  (currently default in production)
==References==
:<math forcemathmode="png">E=mc^2</math>


*{{citation | first =| last = E. Bonan, | title = Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)| journal = C. R. Acad. Sci. Paris | volume =262| year = 1966  | pages = 127&ndash;129}}.
'''source'''
*{{citation | last = Bryant | first = R.L. | title = Metrics with exceptional holonomy | journal = Annals of Mathematics | issue = 2 | volume = 126 | year = 1987 | pages = 525&ndash;576 | doi = 10.2307/1971360 | publisher = Annals of Mathematics | jstor = 1971360}}.
:<math forcemathmode="source">E=mc^2</math> -->
*{{citation | last = Bryant | first = R.L. | first2 =  S.M. | last2 = Salamon | title = On the construction of some complete metrics with exceptional holonomy | journal = Duke Mathematical Journal | volume = 58 | year = 1989 | pages = 829&ndash;850}}.
*{{citation | first = R. | last = Harvey | first2 = H.B. | last2 = Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47&ndash;157 | doi = 10.1007/BF02392726}}.
*{{citation | first = D.D. | last = Joyce | title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = Oxford University Press |  isbn = 0-19-850601-5 | year = 2000}}.
*{{citation | first = R.C. | last = McLean | title = Deformations of calibrated submanifolds | journal = Communications in Analysis and Geometry | volume = 6 | year = 1998 | pages = 705&ndash;747}}.
*{{citation | first = Spiro | last = Karigiannis | title = What Is . . . a ''G''<sub>2</sub>-Manifold? | journal = AMS Notices | volume = 58 | issue = 04 | pages = 580–581 | year = 2011
    | url = http://www.ams.org/notices/201104/rtx110400580p.pdf }}.


[[Category:Differential geometry]]
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
[[Category:Riemannian geometry]]
[[Category:Structures on manifolds]]


[[sv:G2-mångfald]]
==Demos==
 
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** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
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==Test pages ==
 
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==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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Registered users will be able to choose between the following three rendering modes:

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