arXiv research

We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the beh…

New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.

Study compact plane waves, showing they are essentially standard.

Mathematical treatment of plane waves, proving their inextendibility and completeness.

The paper classifies tensors on specific Lorentzian metrics.

We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on $2$-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classic…

The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.

Classifies 1-connected Lorentzian manifolds with essential conformal groups.

Study of Lorentzian manifolds with specific transformations.

We study the class $\mathcal{K}_{2}+\mathcal{K}_{4}$ of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure is obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is …

We show that generalized plane wave manifolds are complete, strongly geodesically convex, Osserman, Szabo, and Ivanov-Petrova. We show their holonomy groups are nilpotent and that all the local Weyl scalar invariants of these manifolds vanish. We construct isometry invariants on certain families of these manifolds whic…

We show that a Lorentzian homogeneous space admitting a homogeneous structure of type T1 + T3 is either a (locally) symmetric space or a singular homogeneous plane wave.

Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…

We exhibit a family of generalized plane wave manifolds of signature (2,2). The geodesics in these manifolds extend for infinite time (i.e. they are complete), they are spacelike and timelike Jordan Osserman, and they are spacelike and timelike Jordan Ivanov-Petrova. Some are irreducible symmetric spaces. Some are homo…

New Galilean spacetimes found as pp-wave reductions.

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

Study finds rigid submanifolds in spacelike waves under specific conditions.

We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically complete if it is homogeneous, or has constant curvature, or admits a time-like…

New metrics constructed dual to specific wave-like geometries.

We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on …

Recent advances in twistor theory are applied to geometric optics in ${\Bbb{R}}^3$. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming w…

Extends Penrose limit to Finsler spacetimes.

Penrose limit results for specific 3-surfaces in space-time.

Departing from the observation that the Penrose limit of AdS_3 x S^3 is a group contraction in the sense of Inonu and Wigner, we explore the relation between the symmetric D-branes of AdS_3 x S^3 and those of its Penrose limit, a six-dimensional symmetric plane wave analogous to the four-dimensional Nappi--Witten space…

New metrics link Kähler and pp-wave spacetimes.

A discrete method approximates hyperbolic curvature flow in the plane.

The notion of a causal boundary for a spacetime has been a controversial topic during the last three decades. Moreover, recently the role of the boundary in the AdS/CFT correspondence for plane waves, have stimulated its redefinition with some possible alternatives. Our aim is threefold. First, to review the different …

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…

Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that th…

Geometric flow on curves in S^3 generates YO equations solutions.

New metrics derived from geodesics simplify semi-Riemannian geometry.

Impulsive waves contradict a 1962 conjecture about pp-waves.

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connected…

Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry an…

Unified cosmological and Einstein polytope theories.

Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.

Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…

It is well known that Lagrangian dynamical systems naturally arise in describing wave front dynamics in the limit of short waves (which is called pseudoclassical limit or limit of geometrical optics). Wave fronts are the surfaces of constant phase, their points move along lines which are called rays. In non-homogeneous…

In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obt…

We show that the conformal Penrose limit is an ordinary plane wave limit in a higher dimensional framework which resolves the spacetime singularity. The higher dimensional framework is provided by Ricci-flat manifolds which are of the form M_D = M_d x B, where M_d is an Einstein spacetime that has a negative cosmologic…

The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave…

We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A \em{Lagrangian} Engel structure is an Engel 2-plane field on a symplectic 4-manifold for which the 2-planes are Lagrangian with respect to the symplec…

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

Study K-theory of homogeneous spaces of Lie groups.

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…

We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…