Abstract
AbstractWe 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.
Funder
Narodowe Centrum Nauki
Grantová Agentura České Republiky
Consejo Nacional de Ciencia y Tecnología
Publisher
Springer Science and Business Media LLC
Reference31 articles.
1. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambitoric geometry I: Einstein metrics and extremal ambikähler structures. J. Reine Angew. Math. 721, 109–147 (2016)
2. Akivis, M.A., Goldberg, V.V.: Conformal differential geometry and its generalizations. Pure and Applied Mathematics (New York). Wiley, New York (1996)
3. An, D., Nurowski, P.: Twistor space for rolling bodies. Comm. Math. Phys. 326(2), 393–414 (2014)
4. An, D., Nurowski, P.: Symmetric $$(2,3,5)$$ distributions, an interesting ODE of 7th order and Plebański metric. J. Geom. Phys. 126, 93–100 (2018)
5. Bergery, L.B., Ikemakhen, A.: Sur l’holonomie des variétés pseudo-riemanniennes de signature $$(n, n)$$. Bull. Soc. Math. France 125(1), 93–114 (1997)
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献