arXiv research

Classifies Heisenberg-invariant self-dual Einstein manifolds with explicit metrics.

The paper classifies metrics on a Heisenberg group's cotangent bundle.

While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses:…

Study classifies gradient almost Ricci solitons in Lorentzian and neutral signatures.

It is shown that if a compact four-dimensional manifold with metric of neutral signature is Jordan-Osserman, then it is either of constant sectional curvature or Ricci flat.

The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective co…

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possib…

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither sa…

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined …

Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generali…

In this paper we construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. Curvature of these manifolds is studied in details. These are obtained by the para-quaternionic reduction. We compare our examples with the orbifolds $\oo$ given by Galic…

We study the isometry groups and Killing vector fields of a family of pseudo-Riemannian metrics on Euclidean space which have neutral signature (3+2p,3+2p). All are p+2 curvature homogeneous, all have vanishing Weyl scalar invariants, all are geodesically complete, and all are 0-curvature modeled on an indecomposible s…

In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jo…

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an an…

Let $M$ be an $n$-dimensional manifold with a torsion free affine connection $\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider some of the geometrical aspect of a twisted Riemannian extension which provide a link between the affine geometry of $(M,\nabla)$ and the neutral signature pseudo-Riem…

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss t…

We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal…

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …

Given a projective structure on a surface $N$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space $M$ of a certain rank $2$ affine bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature and admits …

We describe the local structure of self-dual gradient Ricci solitons in neutral signature. If the Ricci soliton is non-isotropic then it is locally conformally flat and locally isometric to a warped product of the form $I\times_\varphi N(c)$, where $N(c)$ is a space of constant curvature. If the Ricci soliton is isotro…

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla$ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is called natural Riemann extension and it is a generalization (made by M. Sekizaw…

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space ${\Bbb{T}}$ of oriented affine lines in ${\Bbb{R}}^3$ with the tangent bundle of $S^2$. Thus, the round metric on $S^2$ induces a Kähler structure on ${\Bbb{T}}$ which turns out to h…

Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures $(M,[g])$ with a null conformal Killing vector. We show that $M$ is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural pr…

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel …

A topological defect separating a pair of two-dimensional CFTs is a codimension one interface along which all components of the stress-energy tensor glue continuously. We study topological defects of the bosonic, (0,1)- and (0,2)-supersymmetric sigma models in two dimensions. We find a geometric classification of such …

We construct a Kähler structure (${\mathbb{J}},Ω,{\mathbb{G}}$) on the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space ${\mathbb{H}}^3$ and investigate its properties. We prove that (${\mathbb{L}}({\mathbb{H}}^3),{\mathbb{J}})$ is biholomorphic to ${\mathbb{P}}^1\times{\mathbb{P}}^1-\ba…

We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product const…

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov…

In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G* are complex Lie groups. We also prove that a He…

Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …

For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature i…

Study of pure spinors on neutral manifolds with applications to supersymmetric solutions.

Let $(Σ_1,g_1)$ and $(Σ_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures $(G^ε,J,Ω^ε)$ on the product 4-manifold $Σ_1\timesΣ_2$ given by $G^ε=g_1\oplus εg_2$, $ε=\pm 1$ and $J$ is the canonical product complex structure. Thus for $ε=1$ the Kähler metr…

The paper connects web theory to heavenly PDEs and Einstein metrics.

The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric deformation of a surface in euclidean three space. This interpretation is a differen…

I apply the algebraic classification of self-adjoint endomorphisms of ${\bf R}^{2,2}$ provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar re…

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their…

We review the subject of four dimensional anti-self-dual conformal structures with signature (+ + - -). Both local and global questions are discussed. Most of the material is well known in the literature and we present it in a way which underlines the connection with integrable systems. Some of the results - e.g. the L…

Given an oriented Riemannian surface $(Σ, g)$, its tangent bundle $TΣ$ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $TΣ$ which are both Lagr…

Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds of signature (2s,s) for any s (which is at least 2) which are spacelike Jordan …

It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $Ω^*$. If $(\M,\j,g,ω)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T\M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(\J,\G,…

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems…

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. I…

The paper extends a theorem for complex structures on Lie groups to Courant algebroids.

Affine manifolds linked to integrable equations and geometric structures.

This thesis surveys various metrics on Riemann surface spaces.