arXiv research

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Si…

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ satisfies a vector differential equation of fourth order. The general so…

We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-…

We relate in this note the classical Schwarzian derivative to the curvature of time-like curves in Lorentz surfaces of constant curvature.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Study of longest arcs and cut loci in deformed anti de-Sitter spaces.

We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit specia…

We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural no…

The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.

Investigates physical properties on surfaces of rotation using Clairaut's theorem.

Study links Hopf differentials to curvature line flows on time-like CMC surfaces.

In this letter, two explicit self-similar solutions to a graph representation of time-like extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$ are given. Meanwhile, there is an untable eigenvalue in the linearized time-like extremal hypersurfaces equation around two explicit self-similar solutions.

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface …

Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…

Study loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.

The paper studies sections of time-like twistor spaces with specific covariant derivatives.

The study defines invariants for time-like surfaces with real asymptotic lines.

We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached.

The paper classifies time-like surfaces in a static space-time.

Study on surfaces in neutral space forms with zero mean curvature.

Paper proves a new criterion for time-like geodesics in flat spacetimes.

The paper studies umbilics on surfaces in Lorentz-Minkowski space.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped with almost contact B-metric structures. The constructed manifolds are characteri…

Unified estimates for mean curvature in Lorentz-Minkowski space.

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

The parametrization theorem is derived in a flat nD pseudo-complex affine space. The pseudo-complex hyperbolic space accomodates n-number of uncompactified time-like extra dimensions with sugnature (s,r), where s and r are the numbers of minus and plus signs associated with the diagonalized metric matrix. The main resu…

Study Born-Infeld solitons and solve Björling problem for them.

Study classifies translating solitons in Minkowski 3-space, revealing singularities.

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $π$ along time-like curves), the complement of the convex core in $N$ admits a unique foliation by constant Gauss curvature surfaces. This extends, and …

We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence (namely, the spatial metric tensor, the extrinsic curvature tensor and the Riema…

We study the neutral Kähler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metr…

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…

Killing vector fields of a closed homogeneous and isotropic universe are studied. It is shown that in general case there is no time-like Killing vector fields in such a universe. Two exceptional cases are revealed.

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_ξ(S)$. The definition of $L^2_ξ(S)$ involves the choice of a $p$-dimensional time-like subbundle $ξ\subset TM$. We establish a sufficient criterion for …

New approach constructs symplectic structure on pseudo-Riemannian geodesics.

We study the six-dimensional pseudo-Riemannian spaces with two time-like coordinates that admit non-homothetic infinitesimal projective transformations. The metrics are manifestly obtained and the projective group properties are determined. We also find a generic defining of projective motion in the 6-dimensional rigid…

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

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvatur…

Let $M$ be a globally hyperbolic maximal compact $3$-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that $M$ admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularit…

In the framework of standard static space times, we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories. We concentrate our study on questions about the sub-harmonicity of the warping function, the scalar curvature …

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a gen…

A type of almost contact hypersurfaces with Norden metric of a Kaehler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kaehler curvature tensor is obtained. Som…

Geodesic completeness proven for certain Lorentzian spaces.

Extended recurrent pseudo-Riemannian manifolds were introduced by Mileva Prvanovic'. We reconsider her work in the light of recent results and show that the manifold is conformally flat, and it is a space of quasi-constant curvature. We also show that an extended recurrent Lorentzian manifold, with time-like associated…