arXiv research

Timelike geometry of spherical simplices is shown to be isometric to vector spaces.

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

The aim of this paper is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. Involute trajectory ruled surfaces generated by the Frenet trihedron, moving along spacelike involutes of a given timelike space curve, is stated according to Lorentzian timelike angle …

The study classifies timelike meridian surfaces in Minkowski 4-space.

Isophote comprises a locus of the surface points whose normal vectors make a constant angle with a fixed vector. In this paper, isophote curves are studied on timelike surfaces in Minkowski 3-space E31. The axises of spacelike and timelike isophote curves are found via their Darboux frames. Subsequently, the relationsh…

New curvature obstruction for Killing vector fields on Lorentzian manifolds.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is …

Study explores geometric implications of timelike conformal Killing vectors.

We study Finsler spacetimes and Killing vector fields taking care of the fact that the generalized metric tensor associated to the Lorentz-Finsler function $L$ is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vect…

Rigidity results for hypersurfaces in warped spacetimes.

Study of timelike surfaces in Minkowski space with specific geometric properties.

A function that optimally aligns a timelike vector field with its gradients

In this paper, space and timelike admissible Smarandache curves in the pseudo- Galilean 3-space are investigated. Also, Smarandache curves of the position vector of space and timelike arbitrary curve and some of its special curves in the pseudo- Galilean 3-space are obtained. To confirm our main results, some examples …

We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…

In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet frames of these timelike curves are not the same. We have moved the director vector …

The paper studies spacelike curves and timelike ruled surfaces in Minkowski space.

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some n…

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a t…

Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.

The aim of this paper is to determine criteria of being integral curve for the geodesic spray of the natural lift curves of the spherical indicatrices of the involutes of a given spacelike curve with a timelike binormal in Minkowski 3-space. Furthermore, some interesting results about the spacelike evolute curve with t…

Proves that timelike maximal surfaces in (1+2) Minkowski space are always embedded.

The paper studies timelike minimal surfaces in De Sitter space using complex analysis.

Study finds limits on curvature and shape of certain spacetimes.

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…

Characterizes spacetimes using doubly torqued vectors.

We consider a unit speed timelike curve $α$ in Minkowski 4-space $E_1^4$ and denote the Frenet frame of $α$ by $\{T,N,B_1,B_2\}$. We say that $α$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $E_1^4$. In this work we study those hel…

Given a constant vector field $Z$ in Minkowski space, a timelike surface is said to have a canonical null direction with respect to $Z$ if the projection of $Z$ on the tangent space of the surface gives a lightlike vector field. In this paper we describe these surfaces in the ruled case. For example when the Minkowski …

Isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz-Minkowski space and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give so…

Formally constructs metrics near timelike geodesics in vacuum spacetimes.

The paper classifies rotational K^α-translators in Minkowski space.

We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corres…

We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $γ$ in such a torus is said to be homologically maximizing if one (hence every) lift of $γ$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics…

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.

The study defines conditions for Finsler spacetime structures in $(α,β)$-metrics and identifies their isometries.

In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …

We study the existence of a non-spacelike isometry, ζ, in higher dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of constraints for the metric functions in each case. Within the class of N dimensional CSI Ku…

We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are determined up to a motion by eight invariant functions satisfying some natural conditions. The subclass of Chen surfaces is char…

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formul…

New loops found in universe's timeline, challenging traditional time direction.

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standar…

In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the M…

Complete description of flat Lorentzian Lie groups solved.

A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder…

Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate insta…

Paper classifies timelike Bonnet surfaces in Lorentzian 3-manifolds.