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| {{DISPLAYTITLE:G<sub>2</sub> manifold}}
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| 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 [[special orthogonal group]] SO(7) that preserves a [[spinor]] in the eight-dimensional [[spinor representation]] or lastly as the subgroup of the [[general linear group]] GL(7) which preserves the non-degenerate 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. | |
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| == 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]],
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| * a [[spin manifold]].
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| == 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.
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| ==See also==
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| *[[Spin(7)-manifold]]
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| *[[Calabi–Yau manifold]]
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| == Connections to physics ==
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| 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.
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| ''See also'': [[Calabi-Yau manifold]], [[Spin(7) manifold]]
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| ==References==
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| *{{citation | first =| last = E. Bonan, | authorlink=Edmond 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 = 127–129}}.
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| *{{citation | last = Bryant | first = R.L. | title = Metrics with exceptional holonomy | journal = Annals of Mathematics | issue = 2 | volume = 126 | year = 1987 | pages = 525–576 | doi = 10.2307/1971360 | publisher = Annals of Mathematics | jstor = 1971360}}.
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| *{{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 = 829–850}}.
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| *{{citation |last1 =M. Fernandez| last2 =A. Gray | title = Riemannian manifolds with structure group G2| journal = Ann. Mat.Pura Appl. | volume =32| year =1982| pages = 19–845 }}.
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| *{{citation | first = R. | last = Harvey | first2 = H.B. | last2 = Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47–157 | doi=10.1007/BF02392726}}.
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| *{{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}}.
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| *{{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
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| | url = http://www.ams.org/notices/201104/rtx110400580p.pdf }}.
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| [[Category:Differential geometry]]
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| [[Category:Riemannian geometry]]
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| [[Category:Structures on manifolds]]
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