Quaternion-Kähler manifold

From Wikipedia, the free encyclopedia

In differential geometry, quaternion-Kähler manifold (a.k.a. quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy is reduced to $Sp(n)\cdot Sp(1)$ .

Another, more explicit, definition, uses a 4-dimensional sub-algebra $H\subset End(TM)$ of endomorphisms of a tangent bundle to a Riemannian $M$ . For $M$ to be quaternion-Kähler, $H$ should be preserved by the Levi-Civita connection and locally isomorphic to quaternions, in such a way that unitary quaternions $h\in H$ act on $T M$ preserving the metric.

Notice that this definition includes hyperkähler manifolds (see the next section).

1 Ricci curvature
2 Examples
3 Twistor spaces
4 Reference

[edit] Ricci curvature

Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies as the only manifolds of special holonomy with non-trivial 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 $Sp(n)\cdot Sp(1)$ and with non-trivial Ricci curvature (which is constant).

Quaternion-Kähler manifolds are split naturally among those with positive and negative Ricci curvature.

[edit] Examples

There are no examples of complete quaternion-Kähler manifolds which are not locally symmetric. Symmetric quaternion-Kähler manifolds are known as Wolf spaces. For any simple Lie group $G$ , there is a unique Wolf space $G / H$ obtained as a quotient of $G$ by a subgroup $H = H_1 \times SU(2)$ . Here, $S U (2)$ is the $S L (2)$ -triple associated with the highest root of $G$ , and $H 1$ its centralizer in $G$ . The Wolf spaces with positive Ricci curvature are compact and simply connected.

If $G$ is $S U (n)$ , the corresponding Wolf space is the quaternionic projective space $\Bbb H P^n$ . It can be identified with a space of quaternionic lines in $\Bbb H^{n+1}$ .

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 $H\subset End(TM)$ the corresponding 4-dimensional subalgebra, locally isomorphic to quaternions. Consider the corresponding $S 2$ -bundle $S\subset H$ of all $h\in H$ satisfying $h 2 = - 1$ . The points of $S$ are identified with the complex structures on its base. From this, it is apparent that the total space $T w (M)$ of $S$ is equipped with an almost complex structure.

Salamon proved that this almost complex structure is integrable, hence $T w (M)$ is a complex manifold. When the Ricci curvature of $M$ is positive, $T w (M)$ 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] Reference

[1] Salamon, S., Quaternionic Kähler manifolds, Inv. Math. {\bf 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.

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Categories: Differential geometry | Structures on manifolds | Riemannian geometry

Quaternion-Kähler manifold

From Wikipedia, the free encyclopedia

Contents

[edit] Ricci curvature

[edit] Examples

[edit] Twistor spaces

[edit] Reference

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