Einstein-Weyl Spaces Research Articles

Parallel spinors on Lorentzian Weyl spaces

Holonomy algebras of Lorentzian Weyl spin manifolds with weighted parallel spinors are found. For Lorentzian Weyl manifolds admitting recurrent null vector fields are introduced special local coordinates similar to Kundt and Walker ones. Using that, the local form of all Lorentzian Weyl spin manifolds with weighted parallel spinors is given. The Einstein-Weyl equation for the obtained Weyl structures is analyzed and examples of Einstein-Weyl spaces with weighted parallel spinors are given.

Einstein manifolds with torsion and nonmetricity

Manifolds endowed with torsion and nonmetricity are interesting both from the physical and the mathematical points of view. In this paper, we generalize some results presented in the literature. We study Einstein manifolds (i.e., manifolds whose symmetrized Ricci tensor is proportional to the metric) in d dimensions with nonvanishing torsion that has both a trace and a traceless part, and analyze invariance under extended conformal transformations of the corresponding field equations. Then, we compare our results to the case of Einstein manifolds with zero torsion and nonvanishing nonmetricity, where the latter is given in terms of the Weyl vector (Einstein-Weyl spaces). We find that the trace part of the torsion can alternatively be interpreted as the trace part of the nonmetricity. The analysis is subsequently extended to Einstein spaces with both torsion and nonmetricity, where we also discuss the general setting in which the nonmetricity tensor has both a trace and a traceless part. Moreover, we consider and investigate actions involving scalar curvatures obtained from torsionful or nonmetric connections, analyzing their relations with other gravitational theories that appeared previously in the literature. In particular, we show that the Einstein-Cartan action and the scale invariant gravity (also known as conformal gravity) action describe the same dynamics. Then, we consider the Einstein-Hilbert action coupled to a three-form field strength and show that its equations of motion imply that the manifold is Einstein with totally antisymmetric torsion.

Einstein–Weyl spaces and near-horizon geometry

We show that a class of solutions of minimal supergravity in five dimensions is given by lifts of three-dimensional Einstein–Weyl structures of hyper-CR type. We characterise this class as most general near-horizon limits of supersymmetric solutions to the five-dimensional theory. In particular we deduce that a compact spatial section of a horizon can only be a Berger sphere, a product metric on or a flat three-torus.We then consider the problem of reconstructing all supersymmetric solutions from a given near-horizon geometry. By exploiting the ellipticity of the linearised field equations we demonstrate that the moduli space of transverse infinitesimal deformations of a near-horizon geometry is finite-dimensional.

Twistor theory for co-CR quaternionic manifolds and related structures

In a general and non-metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kähler manifolds.

Gauduchon-Tod structures,Simholonomy and De Sitter supergravity

Solutions of five-dimensional De Sitter supergravity admitting Killing spinors are considered, using spinorial geometry techniques. It is shown that the solutions are defined in terms of a one parameter family of 3-dimensional constrained Einstein-Weyl spaces called Gauduchon-Tod structures. They admit a geodesic, expansion-free, twist-free and shear-free null vector field and therefore are a particular type of Kundt geometry. When the Gauduchon-Tod structure reduces to the 3-sphere, the null vector becomes recurrent, and therefore the holonomy is contained in Sim(3), the maximal proper subgroup of the Lorentz group SO(4,1). For these geometries, all scalar invariants built from the curvature are constant. Explicit examples are discussed.

A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence

We construct infinitely many Einstein-Weyl structures on \({S^2\times\mathbb{R}}\) of signature (− + +) which is sufficiently close to the model case of constant curvature, and on which the space-like geodesics are all closed. Such a structure is obtained as a parameter space of a family of holomorphic disks which is associated to a small perturbation of the diagonal of \({\mathbb{CP}^1\times\overline{\mathbb{CP}^1}}\) . The geometry of constructed Einstein-Weyl spaces is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article.

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type

HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can in principle be constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures.

Einstein–Weyl spaces and dispersionless Kadomtsev–Petviashvili equation from Painlevé I and II

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlev\'e transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-K\"ahler metrics.

Selfdual Einstein Metrics with Torus Symmetry

It is well-known that any 4-dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on • 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kähler quotients of • Pm—1 by an (m — 2)-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kähler quotients of • P2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the ( n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G: • there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric; • in that gauge, a simple constraint on the conformal scalar curvature holds; • a non-exact Einstein–Weyl structure may exist only if the ( n−1)-dimensional homogeneous space G/ H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H; • the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′. The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint. We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the ( n−2)-dimensional space G/( H× H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU( n/2) being, thanks to our results, invariant under U(1)× SU( n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.

Applications of Riemannian and Einstein–Weyl Geometry in the Theory of Second-Order Ordinary Differential Equations

We consider some properties of four-dimensional Riemannian spaces whose metric coefficients are associated with the coefficients of second-order nonlinear differential equations, and we study the properties of three-dimensional Einstein–Weyl spaces related to the dual equations b″ = g(a,b,b′), where the function g(a,b,b′) satisfies a special partial differential equation.

The geometry of the Toda equation

I show that solutions of the SU(∞) Toda field equation generating a fixed Einstein–Weyl space are governed by a linear equation on the Einstein–Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein–Weyl space are found. I also give a classification of Einstein–Weyl spaces arising from the Toda equation in more than one way. This classification coincides with a class of spaces found by Ward and hence clarifies some of their properties. I end by discussing the simplest examples.

Einstein–Weyl spaces and third-order differential equations

The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, “On the null surface formalism,” Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., “Null surfaces formation in 3D,” J. Math Phys. (submitted)] are extended to describe Einstein–Weyl spaces, following Cartan [E. Cartan, “Les espaces généralisées et l’integration de certaines classes d’equations différentielles,” C. R. Acad. Sci. 206, 1425–1429 (1938); “La geometria de las ecuaciones diferenciales de tercer order,” Rev. Mat. Hispano-Am. 4, 1–31 (1941)]. In the resulting formalism, Einstein–Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein–Weyl spaces are given.

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new Einstein-Weyl spaces, which we call Einstein-Weyl “with a geodesic symmetry”, give rise to hypercomplex structures with two commuting triholomorphic vector fields.

Compact Einstein-Weyl four-dimensional manifolds

We look for four-dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit four-parameter expression of the distance obtained in a previous work for non-conformally Einstein Einstein-Weyl structures, we show that only four one-parameter families of regular metrics exist on orientable manifolds: they are all of Bianchi type IX and locally conformally Kähler; moreover, in agreement with general results, they have a positive-definite conformal scalar curvature. In a Gauduchon gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann: our simpler parametrization allows us to correct some of their assertions.

Einstein-Weyl structures and Bianchi metrics

We analyse in a systematic way the (non-)compact four-dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally Kähler; in the non-exact Einstein-Weyl case with a Bianchi metric of the type , VIII or IX, we show that the distance may be taken in a diagonal form and we obtain its explicit 4-parameters expression. This extends our previous analysis, limited to the diagonal, Kähler Bianchi IX case.

(4,0) and (4,4) sigma models with a tri-holomorphic Killing vector

We analyze (4,0) supersymmetric σ-models on a four dimensional target space which possess one tri-holomorphic Killing vector which is also assumed to leave invariant the torsion. The problem is reduced to two stages: first finding “special” three dimensional Einstein-Weyl spaces, and second solving a monopole-like equation on the special Einstein-Weyl space. A new class of examples is constructed using as Einstein-Weyl geometry the Berger sphere which includes the round three sphere as a particular case. When the Einstein-Weyl geometry is taken to be the round three sphere we show that the corresponding (4,0) geometries can be lifted to (4,4) geometries with two sets of non-commuting hyper-complex structures.

Three Dimensional Einstein-Weyl Geometry

This paper is an investigation of various differential geometric properties of three-dimensional Einstein-Weyl spaces. We show that, in a suitable gauge and coordinate system, these spaces are analytic. We find classes of explicit examples and we consider the behaviour of godesics in general and in some of these examples. Although most of the material is orginal, our intention is to be expository, and to compare and contrast the theory of Einstein-Weyl spaces with the theory of Einstein spaces.

Minitwistor spaces and Einstein-Weyl spaces

The authors relate the Hitchin correspondence (1982) between minitwistor spaces and Einstein-Weyl spaces to the Penrose correspondence between twistor spaces and spacetimes with anti-self-dual Weyl tensor. They show explicitly how to obtain an Einstein-Weyl space from such a spacetime with a conformal Killing vector and provide some examples. They also relate the converse problem to the solution of a generalised monopole equation.