Quaternion-Kähler manifold

In mathematics quaternionic Kähler manifolds are a research field of differential geometry.

Definition

A connected orientable Riemannian manifold of dimension is a quaternionic Kählermannigfaltigkeit if its holonomy group is included. In case one requires additionally that it is a self-dual Einstein manifold.

Here are the ( compact ) symplectic group and acts on by left multiplication and right multiplication of (as diagonal matrices ) of, which is seen as a subset of.

A quaternionic Kählermannigfaltigkeit is called positive or negative if the Riemannian metric is complete and positive or negative scalar curvature has.

Properties

• A quaternionic Kählermannigfaltigkeit hyperkähler if and only if its scalar curvature vanishes.

• Any positive quaternionic Kählermannigfaltigkeit is compact and simply connected.

• The only quaternionic Kählermannigfaltigkeit with positive sectional curvature is the quaternionisch - projective space.

All known examples are quaternionischer Kähler Wolf spaces. The LeBrun - Salamon conjecture states that all positive quaternionic Kähler symmetric spaces and thus ( according to the classification of symmetric spaces ) are in particular Wolf spaces. ( For n = 1 the conjecture of Hitchin and for n = 2 by Poon - Salamon has been proven. )

Twistor space

At any quaternionic Kählermannigfaltigkeit you associate a so-called " twistor space " as follows. is superimposed on two occasions and locally can be the bundle to a bundle lift. The effect on can be used to locally define an associated quaternionisches line bundle. Even if this does not have to be defined globally, its complex Projektivisierung is certainly globally defined and obtained a bundle

The space is called twistor space of the quaternionic Kählermannigfaltigkeit.

Example: The twistor space of quaternionisch - projective space is the complex projective space and the bundle

Is the canonical projection map.

Set ( LeBrun - Salamon ): The twistor space of a positive quaternionic Kählermannigfaltigkeit is a Fano contact manifold, also compact, simply connected, and Kählersch Einsteinsch.

Furthermore, a positive quaternionic Kählermannigfaltigkeit is exactly then a symmetric space if you twistor space is a homogeneous ( under biholomorphic pictures ) space.