论文标题
Para-Kähler-Einstein 4-manifolds和不可融合的扭曲器分布
Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions
论文作者
论文摘要
We study the local geometry of 4-manifolds equipped with a \emph{para-Kähler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated \emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes.对于具有不变的爱因斯坦常数的PKE指标，该扭曲器分布完全具有两个积分叶子，并且在其补体上是“最大不可汇聚”的，即所谓的（2,3,5）分布。我们的主要结果建立了具有非变化的爱因斯坦常数的PKE度量的反自动双重WEYL张量与相关扭曲器分布的曲折四重奏之间的对应关系。随后将讨论通用分配签名指标的这种对应关系，该指标被证明更多地参与其中。我们使用卡坦的等效方法来生成许多具有非变化爱因斯坦常数的PKE指标的明确示例，其反自我的Weyl Tensor具有特殊的真实Petrov类型。对于真正的petrov型$ D，$，我们获得了完整的本地分类。结合主要结果，这会产生扭曲的分布，其曲折的四分之一具有与构造的PKE指标的PETROV类型相同的代数类型。以类似的方式，可以获得具有任意代数类型的Cartan四分之一的曲折分布。作为我们的PKE示例的副产品，我们自然会在五个维度上获得Para-Sasaki-Einstein指标。此外，我们研究了与某些类别的PKE 4维指标自然相关的各种卡坦几何形状。我们观察到，在某些几何区分的情况下，相应的\ emph {Cartan Connections}满足Yang-Mills方程。然后，我们提供了此类阳米尔斯漫画连接的明确示例。
We study the local geometry of 4-manifolds equipped with a \emph{para-Kähler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated \emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable' on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-vanishing Einstein constant and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan's method of equivalence to produce a large number of explicit examples of pKE metrics with nonvanishing Einstein constant whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type $D,$ we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding \emph{Cartan connections} satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
