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In differential geometry, a ** G_{2} manifold** or

All -manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers.

The fact that might possibly be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this remained consistent with the simplified proof later given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan nonetheless made a useful contribution by showing that,
if such a manifold did in fact exist, it would carry both a parallel 3-form and a parallel 4-form, and that it would necessarily be Ricci-flat.^{[2]}

The first local examples of 7-manifolds with holonomy were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987.^{[3]} Next, complete (but still noncompact) 7-manifolds with holonomy were constructed by Bryant and Simon Salamon in 1989.^{[4]} The first compact 7-manifolds with holonomy were constructed by Dominic Joyce in 1994. Compact manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.^{[5]}

In 2015, a new construction of compact manifolds, due to Alessio Corti, Mark Haskins, Johannes Nordstrőm, and Tommaso Pacini, combined a gluing idea suggested by Simon Donaldson with new algebro-geometric and analytic techniques for constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples.^{[6]}

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 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 supermultiplets equal to the third Betti number of the manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

**^**Harvey, Reese; Lawson, H. Blaine (1982), "Calibrated geometries",*Acta Mathematica*,**148**: 47–157, doi:10.1007/BF02392726, MR 0666108.**^**Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)",*Comptes Rendus de l'Académie des Sciences*,**262**: 127–129.**^**Bryant, Robert L. (1987), "Metrics with exceptional holonomy",*Annals of Mathematics*,**126**(2): 525–576, doi:10.2307/1971360, JSTOR 1971360.**^**Bryant, Robert L.; Salamon, Simon M. (1989), "On the construction of some complete metrics with exceptional holonomy",*Duke Mathematical Journal*,**58**(3): 829–850, doi:10.1215/s0012-7094-89-05839-0, MR 1016448.**^**Joyce, Dominic D. (2000),*Compact Manifolds with Special Holonomy*, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.**^**Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso (2015). "G2-manifolds and associative submanifolds via semi-Fano 3-folds" (PDF).*Duke Mathematical Journal*.**164**(10): 1971–2092. doi:10.1215/00127094-3120743. S2CID 119141666.

- Becker, Katrin; Becker, Melanie; Schwarz, John H. (2007), "Manifolds with G
_{2}and Spin(7) holonomy",*String Theory and M-Theory : A Modern Introduction*, Cambridge University Press, pp. 433–455, ISBN 978-0-521-86069-7. - Fernandez, M.; Gray, A. (1982), "Riemannian manifolds with structure group G
_{2}",*Ann. Mat. Pura Appl.*,**32**: 19–845, doi:10.1007/BF01760975, S2CID 123137620. - Karigiannis, Spiro (2011), "What Is . . . a
*G*_{2}-Manifold?" (PDF),*AMS Notices*,**58**(4): 580–581.