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

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.

Mass partitions via equivariant sections of Stiefel bundles

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of m mass distributions on Rn, the existence of k affinely independent regular q-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a Zq-equivariant section of the Stiefel bundle Vk(Fn) over S(Fn), where Vk(Fn) is the Stiefel manifold of all orthonormal k-frames in Fn, F = R or C, and S(Fn) is the corresponding unit sphere. For example, the parallelizability of RPn when n = 2,4, or 8 implies that any two masses on Rn can be simultaneously bisected by each of (n-1) pairwise-orthogonal hyperplanes, while when q = 3 or 4, the triviality of the circle bundle V2(C2)=Zq over the standard Lens Spaces L3(q) yields that for any mass on R4, there exist a pair of complex orthogonal regular q-fans, each of which equipartitions the mass.

A COUNTEREXAMPLE TO GUILLEMIN'S ZOLLFREI CONJECTURE

We construct Zollfrei Lorentzian metrics on every non-trivial orientable circle bundle over an orientable closed surface. Further we prove a weaker version of Guillemin's conjecture assuming global hyperbolicity of the universal cover.

Skyrmions from gravitational instantons

We propose a construction of Skyrme fields from holonomy of the spin connection of gravitational instantons. The procedure is implemented for Atiyah–Hitchin and Taub–NUT instantons. The skyrmion resulting from the Taub–NUT is given explicitly on the space of orbits of a left translation inside the whole isometry group. The domain of the Taub–NUT skyrmion is a trivial circle bundle over the Poincaré disc. The position of the skyrmion depends on the Taub–NUT mass parameter, and its topological charge is equal to two.

Geometric models of matter

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza–Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and, at least provisionally, the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub–Newman, Unti, Tamburino (Taub–NUT) manifold as a model for the electron, the Atiyah–Hitchin manifold as a model for the proton, with the Fubini–Study metric as a model for the neutron and S 4 with its standard metric as a model for the neutrino.

SOME CONSTRUCTIONS OF ALMOST PARA-HYPERHERMITIAN STRUCTURES ON MANIFOLDS AND TANGENT BUNDLES

In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M × ℝ, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.

Flat circle bundles, pullbacks, and the circle made discrete

We use basic homotopical methods applied to Lie groups made discrete to prove that the real Euler class of a circle bundle vanishes if and only if the bundle is flat.