Lorentzian Plane Research Articles

PLANE KINEMATICS IN LORENTZIAN HOMOTHETIC MULTIPLICATIVE CALCULUS

In this study, Lorentzian plane homothetic multiplicative calculus kinematics is discussed. Lorentzian plane homothetic multiplicative calculus movement, the pole points of a point X relative to the moving and fixed plane are discussed. In this motion, the velocities and accelerations of a point X are obtained. In this motion, the relations between the velocities and accelerations of a point X are obtained. In addition, new theorems and results are given.

Conformal correlators on the Lorentzian torus

The general form of a 2D conformal field theory (CFT) correlator on a Euclidean Riemann surface, Lorentzian plane or Lorentzian cylinder is well known. This paper describes the general form of 2- and 3-point CFT correlators on the Lorentzian torus LT2 which arises as the conformal boundary of the group manifold SL(2,R)≃AdS3/Z. We consider only generic points, thereby omitting an analysis of contact terms, which already exhibits a surprisingly rich structure. The results are relevant to celestial holography, for which the LT2 at the boundary of Klein space is the home of the putative celestial CFT. Published by the American Physical Society 2024

On geometric interpretations of split quaternions

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three‐dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3‐space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).

FIRST AND SECOND ACCELERATION POLES IN LORENTZIAN HOMOTHETIC MOTIONS

In this paper, using matrix methods, we obtained rotation polein one-parameter motion on the Lorentzian plane homothetic motions andpole orbits, accelerations and combinations of accelerations, …rst and secondin acceleration poles. Moreover, some new theorems are given

Characterizations of special time-like curves in Lorentzian plane 𝕃2

In this paper, we first obtain the differential equation characterizing position vector of time-like curve in Lorentzian plane [Formula: see text] Then we study the special curves such as Smarandache curves, circular indicatrices, and curves of constant breadth in Lorentzian plane [Formula: see text] We give some characterizations of these special curves in [Formula: see text]

Shape curvatures of the Lorentzian plane curves

In this paper, we examine the Lorentzian similar plane curvesusing the hyperbolic structure and spherical arc length parameter. We classifyall self-similar Lorentzian plane curves and give formulas for pseudo shapecurvatures of evolute, involute and parallel curves of a nonnull plane curve

Lagrangian H-Umbilical Surfaces in Complex Lorentzian Plane

Lagrangian H-Umbilical Surfaces in Complex Lorentzian Plane

Lorentz-conformal transformations in the plane

While conformal transformations of the plane preserve Laplace’s equation, Lorentz-conformal mappings preserve the wave equation. This property gives rise to a rich Lorentzian geometry in dimension 1+1. In elucidating the geometry of the Lorentzian plane, this work provides a window into the field of pseudo-Riemannian geometry, where the Lorentzian plane is of fundamental importance. In the Lorentzian plane, curvilinear quadrilaterals and pairs of crossing curves are transformed under nonlinear Lorentz-conformal mappings via geometric constructions that can be expressed also by functional formulas. Classes of Lorentz-conformal maps are characterized by their symmetries under subgroups of the dihedral group of order eight, and unfoldings of non-invertible mappings into invertible ones are reflected in a change of the symmetry group. The questions are simple; but the answers are not obvious, yet have beautiful geometric, algebraic, and functional descriptions and proofs. This is due to the simple form of nonlinear Lorentz-conformal transformations in dimension 1+1, provided by characteristic coordinates.

Polygons of the Lorentzian plane and spherical simplexes

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.

Ricci and Yamabe solitons on second-order symmetric, and plane wave 4-dimensional Lorentzian manifolds

We show that a proper second-order symmetric spacetime, and four-dimensional Lorentzian plane wave manifolds admit different vector fields resulting in expanding, steady and shrinking Ricci and Yamabe solitons. Moreover, it is proved that those Ricci and Yamabe solitons are gradient only in the steady case.

Bicomplex number and tensor product surfaces in $ \mathbb{R}_2^4 $

We show that a hyperquadric M in $ \mathbb{R}_2^4 $ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric M. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Moreover, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in $ \mathbb{R}_2^4 $ .

General Boosts in Lorentzian Plane

In this paper we obtained the equations of boosts (rotations in the Lorentzian plane ) about an arbitrary point (h, k) and showed that the set of all boosts is not a group.

Cauchy Formulas for Enveloping Curves in the Lorentzian Plane and Lorentzian Kinematics

In the Lorentzian plane, we give Cauchy-length formulas to the envelope of a family of lines. Using these, we prove the length of the enveloping trajectories of non-null lines under the planar Lorentzian motions and give the Holditch-type theorems for the length of the enveloping trajectories. Furthermore, Holditch-type theorem for the orbit areas of three collinear points which is given by Yuce and Kuruoglu [8] is generalized to three non-collinear points.

Lagrangian minimal surfaces in Lorentzian complex plane

Flat Lagrangian minimal surfaces in the Lorentzian complex plane \({\mathbb{C}}^{2}_{1}\) are classified by B. Y. Chen and L. Vrancken in [8]. On the other hand, Vrancken proves in [11] that Lagrangian minimal surfaces of constant curvature in \({\mathbb{C}}^{2}_{1}\) are flat surfaces. In this article, we classify all Lagrangian minimal surfaces in \({\mathbb{C}}^{2}_{1}\) which are free from flat points.

Classification of Flat Lagrangian Surfaces in Complex Lorentzian Plane

One of the most fundamental problems in the study of Lagrangian submanifolds from Riemannian geometric point of view is to classify Lagrangian immersions of real space forms into complex space forms. The main purpose of this paper is thus to classify flat Lagrangian surfaces in the Lorentzian complex plane \( {\text{\bf {C}}}^{2}_{1} \). Our main result states that there are thirty-eight families of flat Lagrangian surfaces in \( {\text{\bf {C}}}^{2}_{1} \). Conversely, every flat Lagrangian surface in \( {\text{\bf {C}}}^{2}_{1} \) is locally congruent to one of the thirty-eight families.

PROJECTIONS OF PSEUDOSPHERE IN THE LORENTZ 3-SPACE

In this paper, we study the map projections from pseudosphere S2 1 onto the non-lightlike surfaces in the 3-dimensional Lorentzian space, L3, with curvature zero. We show geometrical means and properties of R × S1 1−cylindrical, S1 × L−cylindrical and R × H1 0−cylindrical projections defined on S2 1 to cylinders R × S1 1 , S1 × L and R × H1 0 , respectively, and orthographic and stereographic projections on S2 1 to Lorentzian plane, L2.

On the Napoleon-Torricelli configuration in Affine Cayley-Klein planes

There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ωk-and anti-Ωk-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived.

Hyperbolic Sine and Cosine Rules for Geodesic Triangles on the Hyperbolic Unit Sphere \(H_{0}^{2}\)

In plane Lorentzian geometry it is studied points, timelike, spacelike and lightlike lines, triangles, etc [4]. On the hyperbolic sphere, there are points, but there are no straight lines, at least not in the usual sense. However, straight timelike lines in the Lorentzian plane are characterized by the fact that they are the shortest paths between points. The curves on the hyperbolic sphere with the same property are hyperbolic circles. Thus it is natural to use these circles as replacements for timelike lines. The formulas for the sine and cosine rules are given for the Euclidean sphere 2 S [2, 3, 6] and hyperbolic sphere [5]. In this study, we obtained formulas related with the spacelike angles and hyperbolic angles corresponding to the sides of geodesic triangles on hyperbolic unit sphere \(H_{0}^{2}\).

Support functions and integral formulas in the Lorentzian plane

In the Lorentzian plane, the support function of a closed convex curve is not a periodic one. We redefine this concept and the envelope of a families of lines. Using these, we prove integral formulas in the Crofton's style for convex sets in the Lorentzian plane.

Three Lorentzian planes moving with respect to one another and pole points

In this paper we have considered a Lorentzian plane A moving with respect to the planes L and L’. We have discussed here on the three Lorentzian planes A, L and L’ instead of the two planes in [1 ]. Also, Lorentzian pole points and pole line of the mentioned planes are evaluated at a time “t”.