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| In [[differential geometry]], a '''quaternion-Kähler manifold''' (or '''quaternionic Kähler manifold''') is a Riemannian manifold whose [[holonomy|Riemannian holonomy group]] is a subgroup of Sp(''n'')·Sp(1). Although this definition ''includes'' [[hyperkähler manifold]]s, these are often excluded from the definition of a quaternion-Kähler manifold by imposing the condition that the [[scalar curvature]] is nonzero, or that the holonomy group is equal to Sp(''n'')·Sp(1).
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| The definition introduced by [[Edmond Bonan]]<ref>{{citation| first =Edmond| last =Bonan |authorlink=Edmond Bonan| title =Structure presque quaternale sur une variété différentiable| journal = C. R. Acad. Sci. Paris | volume =261| year = 1965 | pages =5445–5448}}.</ref> in 1965, uses a 3-dimensional subbundle ''H'' of End(T''M'') of endomorphisms of the tangent bundle to a Riemannian ''M'', that in 1976 Stefano Marchiafava and Giuliano Romani called ''I fibrato di Bonan''
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| .<ref>{{citation | last =S.Marchiafava | first =| first2 = | last2 =G.Romani| title = Sui fibrati con struttura quaternionale generalizzata| journal = Annali di Matematica pura ed applicata | volume = 107 | year = 1976 | pages =131–157}}</ref> For ''M'' to be quaternion-Kähler, ''H'' should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions which act on T''M'' preserving the metric. Simultaneously, in 1965, [[Edmond Bonan]] and Vivian Yoh Kraines<ref>{{citation| first =Vivian Yoh | last = Kraines | title = Topology of quaternionic manifolds | journal = Bull. Amer. Math. Soc| volume =71,3, 1 | year = 1965 | pages = 526–527}}.</ref> constructed the parallel 4-form. It was not until 1982 that [[Edmond Bonan]] proved an outstanding result : the analogue of hard Lefschetz theorem <ref> | |
| {{citation |first =| last = E. Bonan, | authorlink=Edmond Bonan| title =Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique|volume =295| journal = C. R. Acad. Sci. Paris, série I|year =1982| pages = 115–118}}.
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| *{{citation | title =<math>\ \ h<n, \ \Lambda^{2n+2h}= \Omega^h\wedge\Lambda^{2n-2h}</math>}}
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| *{{citation | title =<math>\ \ p\leq 2n,\ \phi_{p}=\sum_{0}^{[p/4]}\Omega^{h}\wedge\mu_{p-4h},\ \ \Omega\wedge*\mu_{p-4h}=0</math>}}
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| *{{citation | title =<math> \ \ p<n-1, \ b_{2p}\leq b_{2p+4}</math>}}
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| *{{citation | title =<math> \ \ p\leq n-1, \ b_{2p-1}\leq b_{2p+1}</math>}}
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| *{{citation |first =| last = E. Bonan, | authorlink=Edmond Bonan| title =Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique| journal = C. R. Acad. Sci. Paris, série I|volume =296| year = 1983 | pages = 601–602}}.
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| *{{citation |last1 =T.Nagano| last2 =M.Takeuchi| title =Signature of quaternionic Kaehler manifolds| journal =Proc.Japan Acad. | volume =59| year =1983| pages = 384–386}}.
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| *{{citation | first = Andrew.F. | last = Swann | title =HyperKähler and Quaternionic Kähler Geometry | year = 1990| url = http://home.imf.au.dk/swann/thesisafs.pdf}}.
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| * Edmond Bonan, ''Isomorphismes sur une variété presque hermitienne quaternionique'', Proc. of the Meeting on Quaternionique Structures in Math.and Physics SISSA , Trieste, (1994), 1-6.
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| </ref> for compact Sp(''n'')·Sp(1)-manifold.
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| == Ricci curvature ==
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| Quaternion-Kähler manifolds appear in [[holonomy|Berger's list of Riemannian holonomies]] as the only manifolds of special holonomy with
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| non-zero Ricci curvature. In fact, these manifolds are [[Einstein manifold|Einstein]].If an Einstein constant of a quaternion-Kähler manifold is zero, it is [[hyperkähler manifold|hyperkähler]]. This case is often excluded from the definition. That is, quaternion-Kähler is defined as one with holonomy reduced to Sp(''n'')·Sp(1) and with non-zero Ricci curvature (which is constant).
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| Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.
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| == Examples ==
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| There are no known examples of compact quaternion-Kähler manifolds which are not [[locally symmetric]] or hyperkähler. [[Quaternion-Kähler symmetric space|Symmetric quaternion-Kähler manifolds]] are also known as [[Wolf spaces]]. For any simple Lie group ''G'', there is a unique Wolf space ''G''/''K'' obtained as a quotient of ''G'' by a subgroup
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| :<math> K = K_0 \cdot SU(2). </math>
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| Here, SU(2) is the subgroup associated with the highest root of ''G'', and ''K''<sub>0</sub> is its [[centralizer]] in ''G''. The Wolf spaces with positive Ricci curvature are compact and simply connected.
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| If ''G'' is Sp(''n''+1), the corresponding Wolf space is the '''[[quaternionic projective space]]'''
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| :<math>\mathbb H P^n.</math> | |
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| It can be identified with a space of quaternionic lines in '''H'''<sup>''n''+1</sup>.
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| It is conjectured that all quaternion-Kähler manifolds with positive Ricci curvature are symmetric.
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| == Twistor spaces ==
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| Questions about quaternion-Kähler manifolds of positive Ricci curvature can be translated into the language of algebraic geometry using the
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| methods of ''twistor theory'' (this approach is due to Penrose and Salamon). Let ''M'' be a quaternionic-Kähler manifold, and ''H'' the corresponding subbundle of End(T''M''), pointwise isomorphic to the imaginary quaternions. Consider the corresponding ''S''<sup>2</sup>-bundle
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| ''S'' of all ''h'' in ''H'' satisfying ''h''<sup>2</sup> = -1. The points of ''S'' are identified with the complex structures on its base. Using this, it is can be shown that the total space ''Z'' of ''S'' is equipped with an [[almost complex manifold|almost complex structure]].
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| Salamon proved that this almost complex structure is integrable, hence ''Z'' is a complex manifold. When the Ricci curvature of ''M'' is positive, ''Z'' is a projective [[Fano variety|Fano manifold]], equipped with a holomorphic [[contact manifold|contact structure]].
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| The converse is also true: a projective Fano manifold which admits a holomorphic contact structure is always a twistor space, hence quaternion-Kähler geometry with positive Ricci curvature is essentially equivalent to the geometry of holomorphic contact Fano manifolds.
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| == References ==
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| <references/>
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| *{{citation | last =S.Marchiafava | first =| first2 = | last2 =G.Romani| title = Sui fibrati con struttura quaternionale generalizzata| journal = Annali di Matematica pura ed applicata | volume = 107 | year = 1976 | pages =131–157}}.
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| * Besse, Arthur Lancelot, ''Einstein Manifolds'', Springer-Verlag, New York (1987)
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| * Salamon, Simon, ''Quaternionic Kähler manifolds'', Invent. Math. '''67''' (1982), 143-171.
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| * [[Dominic Joyce]], ''Compact manifolds with special holonomy'', Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.
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| {{DEFAULTSORT:Quaternion-Kahler manifold}}
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| [[Category:Manifolds]]
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| [[Category:Structures on manifolds]]
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| [[Category:Riemannian geometry]]
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| [[de:Quaternionische Kählermannigfaltigkeit]]
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