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@article{2330835,
author = {Bor, Gil and Makhmali, Omid and Nurowski, Pawel},
article_location = {DORDRECHT},
article_number = {1},
doi = {http://dx.doi.org/10.1007/s10711-021-00665-4},
keywords = {Para-complex structure; Para-Kahler structure; Einstein metric; Cartan geometry; Cartan reduction; Petrov type; (2; 3; 5)-distribution; Conformal structure},
language = {eng},
issn = {0046-5755},
journal = {Geometriae Dedicata},
title = {Para-Kahler-Einstein 4-manifolds and non-integrable twistor distributions},
url = {https://link.springer.com/article/10.1007/s10711-021-00665-4},
volume = {216},
year = {2022}
}
TY - JOUR
ID - 2330835
AU - Bor, Gil - Makhmali, Omid - Nurowski, Pawel
PY - 2022
TI - Para-Kahler-Einstein 4-manifolds and non-integrable twistor distributions
JF - Geometriae Dedicata
VL - 216
IS - 1
SP - 1-48
EP - 1-48
PB - Springer
SN - 00465755
KW - Para-complex structure
KW - Para-Kahler structure
KW - Einstein metric
KW - Cartan geometry
KW - Cartan reduction
KW - Petrov type
KW - (2
KW - 3
KW - 5)-distribution
KW - Conformal structure
UR - https://link.springer.com/article/10.1007/s10711-021-00665-4
N2 - We study the local geometry of 4-manifolds equipped with a para-Kahler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated 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 non-zero scalar curvature 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-zero scalar curvature 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 non-zero scalar curvature 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 Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
ER -
BOR, Gil, Omid MAKHMALI and Pawel NUROWSKI. Para-Kahler-Einstein 4-manifolds and non-integrable twistor distributions. \textit{Geometriae Dedicata}. DORDRECHT: Springer, 2022, vol.~216, No~1, p.~1-48. ISSN~0046-5755. Available from: https://dx.doi.org/10.1007/s10711-021-00665-4.
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