Editing Holonomy (section)

===The Berger classification===
In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not [[locally]] a product space) and nonsymmetric (not locally a [[Riemannian symmetric space]]). '''Berger's list''' is as follows:
{| class="wikitable" style="margin: auto;"
! Hol(''g'') || dim(''M'') || Type of manifold || Comments
|-
| [[Special orthogonal group|SO(''n'')]] || ''n'' || [[Orientable manifold]] || —
|-
| [[Unitary group|U(''n'')]] || 2''n'' || [[Kähler manifold]] || Kähler
|-
| [[Special unitary group|SU(''n'')]] || 2''n'' || [[Calabi–Yau manifold]] || [[Ricci-flat]], Kähler
|-
| Sp(''n'') · Sp(1) || 4''n'' || [[Quaternion-Kähler manifold]] || [[Einstein manifold|Einstein]]
|-
| [[Symplectic group|Sp(''n'')]] || 4''n'' || [[Hyperkähler manifold]] || [[Ricci-flat]], Kähler
|-
| [[G2 (mathematics)|G<sub>2</sub>]] || 7 || [[G2 manifold|G<sub>2</sub> manifold]] || [[Ricci-flat]]
|-
| [[Spin group|Spin(7)]] || 8 || [[Spin(7) manifold]] || [[Ricci-flat]]
|}
Manifolds with holonomy Sp(''n'')·Sp(1) were simultaneously studied  in 1965 by [[Edmond Bonan]] and Vivian Yoh Kraines,
who both discovered that such manifolds would necessarily carry a parallel 4-form.

Manifolds with holonomy G<sub>2</sub> or  Spin(7) were first investigated in abstract by [[Edmond Bonan]] in 1966, who classified  the parallel differential forms that such a manifold would carry, and showed that such a  manifold would necessarily be Ricci-flat. However, no examples such  manifolds would actually be constructed for another 30 years. 

Berger's original list also included the possibility of Spin(9) as a subgroup of SO(16). Riemannian manifolds with such holonomy were later shown independently by D. Alekseevski and Brown-Gray to be necessarily locally symmetric, i.e., locally isometric to the [[Cayley plane]] F<sub>4</sub>/Spin(9) or locally flat. See below.) It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. See [[G2 manifold|G<sub>2</sub> manifold]] and [[Spin(7) manifold]].

Note that Sp(''n'') ⊂ SU(2''n'') ⊂ U(2''n'') ⊂ SO(4''n''), so every [[hyperkähler manifold]] is a [[Calabi–Yau manifold]], every [[Calabi–Yau manifold]] is a [[Kähler manifold]], and every [[Kähler manifold]] is [[orientable manifold|orientable]].

The strange list above was explained by Simons's proof of Berger's theorem. A simple and geometric proof of Berger's theorem was given by [[Carlos E. Olmos]] in 2005. One first shows that if a Riemannian manifold is ''not'' a [[locally symmetric space]] and the reduced holonomy acts irreducibly on the tangent space, then it acts transitively on the unit sphere. The Lie groups acting transitively on spheres are known: they consist of the list above, together with 2 extra cases: the group Spin(9) acting on '''R'''<sup>16</sup>, and the group ''T'' · Sp(''m'') acting on '''R'''<sup>4''m''</sup>. Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the [[Cayley projective plane]]), and the second does not occur at all as a holonomy group.

Berger's original classification also included non-positive-definite pseudo-Riemannian metric non-locally symmetric holonomy. That list consisted of SO(''p'',''q'') of signature (''p'', ''q''), U(''p'', ''q'') and SU(''p'', ''q'') of signature (2''p'', 2''q''), Sp(''p'', ''q'') and Sp(''p'', ''q'')·Sp(1) of signature (4''p'', 4''q''), SO(''n'', '''C''') of signature (''n'', ''n''), SO(''n'', '''H''') of signature (2''n'', 2''n''), split G<sub>2</sub> of signature (4, 3), G<sub>2</sub>('''C''') of signature (7, 7), Spin(4, 3) of signature (4, 4), Spin(7, '''C''') of signature (7,7), Spin(5,4) of signature (8,8) and, lastly, Spin(9, '''C''') of signature (16,16). The split and complexified Spin(9) are necessarily locally symmetric as above and should not have been on the list. The complexified holonomies SO(''n'', '''C'''), G<sub>2</sub>('''C'''), and Spin(7,'''C''') may be realized from complexifying real analytic Riemannian manifolds. The last case, manifolds with holonomy contained in SO(''n'', '''H'''), were shown to be locally flat by R. McLean.<ref>{{citation
 | last = Bryant | first = Robert L.
 | contribution = Classical, exceptional, and exotic holonomies: a status report
 | contribution-url = https://ftp.gwdg.de/pub/misc/EMIS/journals/SC/1996/1/pdf/smf_sem-cong_1_93-165.pdf
 | isbn = 2-85629-047-7
 | mr = 1427757
 | pages = 93–165
 | publisher = Soc. Math. France, Paris
 | series = Sémin. Congr.
 | title = Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)
 | volume = 1
 | year = 1996}}</ref>

Riemannian symmetric spaces, which are locally isometric to [[homogeneous space]]s ''G''/''H'' have local holonomy isomorphic to ''H''. These too have been [[list of simple Lie groups|completely classified]].

Finally, Berger's paper lists possible holonomy groups of manifolds with only a torsion-free [[affine connection]]; this is discussed below.