Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

Abstract

We study the local geometry of 4-manifolds equipped with a para-Kähler-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.

Hyperheavenly spaces and their application in Walker and para-Kähler geometries: part II

“…It is Theorem 4.2 and the metric (4.43) which is a general metric of the para-Kähler Einstein spaces for which SD Weyl tensor is algebraically special. The metric (4.43) depends on 4 functions of two variables like it was proved in [3]. The types {[III] e ⊗ [D] nn , [++, ++]} and {[N] e ⊗ [D] nn , [++, ++]} have been also found and they depend on 3 and 2 functions of two variables, respectively 4 .…”

“…0 (4.45)Proof. C(3) = 0 implies A = 0 and C(2) = 0 yields (4.45) (compare (4.39)).Type{[N] e ⊗ [D] nn , [++, ++]} is a little more complicated. Eq.…”

“…To be more precise, solution (3.13) is a metric of the type {[D] ee ⊗ [D] nn , [++, −−, −−, ++]}. It belongs to one of the classes described in[3].…”

“…According to our best knowledge almost all the metrics which we found are new results (see the summary Tables1 and 2). However, it seems that pKE metrics have the most transparent geometrical meaning and via their relation with twistor distributions[3] are physically relevant. Thus, we think that the most important results of the paper are Theorems 3.4 and 4.2.…”