Editing Holonomy (section)

==Riemannian holonomy==
The holonomy of a [[Riemannian manifold]] (''M'', ''g'') is   the holonomy group of the [[Levi-Civita connection]] on the [[tangent bundle]] to ''M''. A 'generic' ''n''-[[dimension]]al [[Riemannian manifold]] has an [[orthogonal group|O(''n'')]] holonomy, or [[special orthogonal group|SO(''n'')]] if it is [[orientable manifold|orientable]].  Manifolds whose holonomy groups are proper subgroups of O(''n'') or SO(''n'') have special properties.

One of the earliest fundamental results on Riemannian holonomy is the theorem of {{harvtxt|Borel|Lichnerowicz|1952}}, which asserts that the restricted holonomy group is a closed Lie subgroup of O(''n'').  In particular, it is [[compact set|compact]].

===Reducible holonomy and the de Rham decomposition===
Let ''x'' ∈ ''M'' be an arbitrary point.  Then the holonomy group Hol(''M'') acts on the tangent space T<sub>x</sub>''M''.  This action may either be irreducible as a group representation, or reducible in the sense that there is a splitting of T<sub>x</sub>''M'' into orthogonal subspaces T<sub>x</sub>''M'' = T′<sub>x</sub>''M'' ⊕ T″<sub>x</sub>''M'', each of which is invariant under the action of Hol(''M'').  In the latter case, ''M'' is said to be '''reducible'''.

Suppose that ''M'' is a reducible manifold.  Allowing the point ''x'' to vary, the bundles T′''M'' and T″''M'' formed by the reduction of the tangent space at each point are smooth distributions which are [[Frobenius integration theorem|integrable in the sense of Frobenius]].  The [[integral manifold]]s of these distributions are totally geodesic submanifolds.  So ''M'' is locally a Cartesian product ''M′'' × ''M″''.  The (local) de Rham isomorphism follows by continuing this process until a complete reduction of the tangent space is achieved:<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=§IV.5}}</ref>

: Let ''M'' be a [[simply connected]] Riemannian manifold,<ref>This theorem generalizes to non-simply connected manifolds, but the statement is more complicated.</ref> and T''M'' = T<sup>(0)</sup>''M'' ⊕ T<sup>(1)</sup>''M'' ⊕ ⋯ ⊕ T<sup>(''k'')</sup>''M'' be the complete reduction of the tangent bundle under the action of the holonomy group.  Suppose that T<sup>(0)</sup>''M'' consists of vectors invariant under the holonomy group (i.e., such that the holonomy representation is trivial).  Then locally ''M'' is isometric to a product
:: <math>V_0\times V_1\times \cdots\times V_k,</math>
: where ''V''<sub>0</sub> is an open set in a [[Euclidean space]], and each ''V<sub>i</sub>'' is an integral manifold for T<sup>(''i'')</sup>''M''.  Furthermore, Hol(''M'') splits as a direct product of the holonomy groups of each ''M<sub>i</sub>'', the maximal integral manifold of T<sup>(''i'')</sup> through a point.

If, moreover, ''M'' is assumed to be [[geodesically complete]], then the theorem holds globally, and each ''M<sub>i</sub>'' is a geodesically complete manifold.<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=§IV.6}}</ref>

===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.

===Special holonomy and spinors===
Manifolds with special holonomy are characterized by the presence  of parallel [[spinor]]s, meaning spinor fields with vanishing covariant derivative.<ref name="Lawson">{{harvnb|Lawson|Michelsohn|1989|loc=§IV.9–10}}</ref>  In particular, the following facts hold:
* Hol(ω) ⊂ ''U''(n) if and only if ''M'' admits a covariantly constant (or ''parallel'') projective pure spinor field.
* If ''M'' is a [[spin structure|spin manifold]], then Hol(ω) ⊂ ''SU''(n) if and only if ''M'' admits at least two linearly independent  parallel pure spinor fields.  In fact, a parallel pure spinor field determines a canonical reduction of the structure group to ''SU''(''n'').
* If ''M'' is a seven-dimensional spin manifold, then ''M'' carries a non-trivial parallel spinor field if and only if the holonomy is contained in G<sub>2</sub>.
* If ''M'' is an eight-dimensional spin manifold, then ''M'' carries a non-trivial parallel spinor field if and only if the holonomy is contained in Spin(7).
<!--Anyone know of nice results specific to the hyper-Kaehler and quaternion-Kaehler holonomies?-->

The unitary and special unitary holonomies are often studied in connection with [[twistor theory]],<ref>{{harvnb|Baum|Friedrich|Grunewald|Kath|1991}}</ref> as well as in the study of [[almost complex structure]]s.<ref name="Lawson" />

===Applications===

====String Theory====
Riemannian manifolds with special holonomy play an important role in [[string theory]] [[compactification (physics)|compactifications]].
<ref>{{citation | author = Gubser, S. | title = Special holonomy in string theory and M-theory | editor=Gubser S.|display-editors=etal}}
+{{citation |  title=Strings, branes and extra dimensions, TASI 2001. Lectures presented at the 2001 TASI school, Boulder, Colorado, USA, 4–29 June 2001.  | place=River Edge, NJ | publisher=World Scientific | isbn=978-981-238-788-2 | pages=197–233 | year=2004 | arxiv=hep-th/0201114 | last1=Gubser | first1=Steven S. }}.</ref>
This is because special holonomy manifolds admit [[Covariance and contravariance of vectors|covariantly]] constant (parallel) [[spinor]]s and thus preserve some fraction of the original [[supersymmetry]]. Most important are compactifications on [[Calabi–Yau manifold]]s with SU(2) or SU(3) holonomy. Also important are compactifications on [[G2 manifold|G<sub>2</sub> manifold]]s.

====Machine Learning====
Computing the holonomy of Riemannian manifolds has been suggested as a way to learn the structure of data manifolds in [[machine learning]], in particular in the context of [[manifold learning]]. As the holonomy group contains information about the global structure of the data manifold, it can be used to identify how the data manifold might decompose into a product of submanifolds. The holonomy cannot be computed exactly due to finite sampling effects, but it is possible to construct a numerical approximation using ideas from [[spectral graph theory]] similar to Vector Diffusion Maps. The resulting algorithm, the Geometric Manifold Component Estimator ({{Smallcaps|GeoManCEr}}) gives a numerical approximation to the de Rham decomposition that can be applied to real-world data.<ref>{{citation |last1=Pfau |first1=David |last2=Higgins |first2=Irina |last3=Botev |first3=Aleksandar |last4=Racanière |first4=Sébastien |title=Disentangling by Subspace Diffusion |journal=Advances in Neural Information Processing Systems |date=2020 |arxiv=2006.12982}}</ref>