Quaternion-Kähler manifold

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In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1).

Another, more explicit, definition, uses a 3-dimensional subbundle H of End(TM) of endomorphisms of the tangent bundle to a Riemannian M. For M to be quaternion-Kähler, H should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions, in such a way that unit imaginary quaternions in H act on TM preserving the metric.

Notice that this definition includes hyperkähler manifolds. However, 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).

Contents 1 Ricci curvature 2 Examples 3 Twistor spaces 4 References

[edit] Ricci curvature

Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies as the only manifolds of special holonomy with non-zero Ricci curvature. In fact, these manifolds are Einstein.

If an Einstein constant of a quaternion-Kähler manifold is zero, it is 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).

Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.

[edit] Examples

There are no examples of compact quaternion-Kähler manifolds which are not locally symmetric or hyperkähler. Symmetric quaternion-Kähler manifolds are 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

.

Here, SU(2) is the subgroup associated with the highest root of G, and K₀ is its centralizer in G. The Wolf spaces with positive Ricci curvature are compact and simply connected.

If G is Sp(n+1), the corresponding Wolf space is the quaternionic projective space

.

It can be identified with a space of quaternionic lines in Hⁿ⁺¹.

It is conjectured that all quaternion-Kähler manifolds with positive Ricci curvature are symmetric.

[edit] Twistor spaces

Questions about quaternion-Kähler manifolds of positive Ricci curvature can be translated into the language of algebraic geometry using the 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(TM), pointwise isomorphic to the imaginary quaternions. Consider the corresponding S²-bundle S of all h in H satisfying h² = -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 structure.

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 manifold, equipped with a holomorphic contact structure.

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.

[edit] References

1. Salamon, S., Quaternionic Kähler manifolds, Inv. Math. 67 (1982), 143-171.
2. Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)
3. Joyce, D., Compact manifolds with special holonomy, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.