Editing G2 manifold (section)

== History ==
The fact that <math>G_2</math> 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 [[James Harris Simons|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.<ref>{{citation | first =Edmond| last = Bonan| author-link=Edmond Bonan| title = Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)| journal = [[Comptes Rendus de l'Académie des Sciences]] | volume =262| year = 1966  | pages = 127&ndash;129}}.</ref>

The first local examples of 7-manifolds with holonomy <math>G_2</math> were finally constructed around 1984 by [[Robert Bryant (mathematician)|Robert Bryant]], and his full proof of their existence  appeared in the Annals in 1987.<ref>{{citation | last = Bryant | first = Robert L. | author-link=Robert Bryant (mathematician)|title = Metrics with exceptional holonomy | journal = [[Annals of Mathematics]] | issue = 2 | volume = 126 | year = 1987 | pages = 525–576 | doi = 10.2307/1971360 | jstor = 1971360}}.</ref> Next, complete (but still noncompact) 7-manifolds with holonomy <math>G_2</math> were constructed by Bryant and Simon Salamon in 1989.<ref>{{citation | last1 = Bryant | first1 = Robert L. |author-link1=Robert Bryant (mathematician)| first2 =  Simon M. | last2 = Salamon | title = On the construction of some complete metrics with exceptional holonomy | journal = [[Duke Mathematical Journal]] | volume = 58 | year = 1989 | issue = 3 | pages = 829–850 | doi = 10.1215/s0012-7094-89-05839-0|mr=1016448 }}.</ref> The first compact 7-manifolds with holonomy <math>G_2</math> were constructed by [[Dominic Joyce]] in 1994. Compact <math>G_2</math> manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.<ref>{{citation | first = Dominic D. | last = Joyce | author-link=Dominic Joyce| title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = [[Oxford University Press]] |  isbn = 0-19-850601-5 | year = 2000}}.</ref>  In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and [[Sema Salur]] that any manifold with a spin structure, and, hence, a <math>G_2</math>-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with <math>G_2</math>-structure.<ref name ="arikanetal">{{citation | first1 = M. Firat | last1 = Arikan | first2 =  Hyunjoo | last2 = Cho | first3 =  Sema | last3 = Salur | author3-link=Sema Salur|title = Existence of compatible contact structures on <math>G_2</math>-manifolds | journal = [[Asian Journal of Mathematics]] | issue = 2 | volume = 17 | year = 2013  | pages = 321–334 | doi = 10.4310/AJM.2013.v17.n2.a3 | arxiv = 1112.2951 | s2cid = 54942812 }}.</ref> In the same paper, it was shown that certain classes of <math>G_2</math>-manifolds admit a contact structure.

In 2015, a new construction of compact <math>G_2</math> 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 manifold]]s with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples.<ref>{{cite journal|last1=Corti|first1= Alessio|author1-link=Alessio Corti|last2= Haskins|first2= Mark|last3= Nordström|first3= Johannes|last4= Pacini|first4= Tommaso |year=2015|title={{math|''G''<sub>2</sub>}}-manifolds and associative submanifolds via semi-Fano 3-folds|journal=[[Duke Mathematical Journal]] |volume=164|issue= 10|pages=1971–2092|doi= 10.1215/00127094-3120743|s2cid= 119141666|url= http://opus.bath.ac.uk/44698/1/g2m_duke_accepted.pdf}} </ref>