Para-K\'ahler-Einstein 4-manifolds and non-integrable twistor distributions
| Autor: | Bor, Gil, Makhmali, Omid, Nurowski, Paweł |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Geometriae Dedicata (2022) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10711-021-00665-4 |
| Popis: | We study the local geometry of 4-manifolds equipped with a \emph{para-K\"ahler-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. Comment: 45 pages; Minor corrections; Final version |
| Databáze: | arXiv |
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