Lightlike Curve Research Articles

Ruled surfaces obtained by curves in the lightlike cone

Abstract In this paper, we consider the ruled surfaces obtained the lightlike curve in the 2-dimensional lightlike cone according to the frame given in [7]. We obtain necessary and sufficient conditions for the ruled surfaces to be minimal and to have a harmonic Gauss map. Then we see that the ruled surface is minimal if and only if it has a harmonic Gauss map.

Lightlike tangent developables in de Sitter 3-space

Developable surfaces are the special ruled surfaces where the Gaussian curvature of each point vanishes. Tangent developable is the most interesting surface among three fundamental types of developable surfaces. This paper investigates lightlike tangent developable generated by a lightlike base curve in de Sitter 3-space. We first give the topological classification of the lightlike curves which lie in de Sitter 3-space by using the theory of finite determinacy. Furthermore, we establish the relationships between lightlike tangent developables and topological types of lightlike curves, and we give the classification of singularities of lightlike tangent developables from the viewpoint of Legendrian singularity theory. Finally, we provide several examples to illustrate our theoretical results.

A new construction on the energy of space curves in unit vector fields in Minkowski space E₂⁴

In this paper, we firstly introduce kinematics properties of a moving particle lying in Minkowski space E₂⁴. We assume that particles corresponds to different type of space curves such that they are characterized by Frenet frame equations. Guided by these, we present geometrical understanding of an energy and pseudo angle on the particle in each Frenet vector fields depending on the particle corresponds to a spacelike, timelike or lightlike curve in E₂⁴. Then we also determine the bending elastic energy functional for the same particle in E₂⁴ by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy on the particle in each Frenet vector field.

Curvature and Torsion Dependent Energy of Elastica and Nonelastica for a Lightlike Curve in the Minkowski Space

Curvature and Torsion Dependent Energy of Elastica and Nonelastica for a Lightlike Curve in the Minkowski Space

Generalized Bertrand Curves in Minkowski 3-Space

We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves.

Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space

UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space and investigate its relationship with the energy of the same curve in Bishop vector fields in . Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space .

Inextensible Flows of Curves on Lightlike Surfaces

In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface.

On Darboux rotation axis of lightlike curves

In this paper, by used Frenet trihedron for a lightlike curve, the motion of Darboux rotation axis is seperated to two simultaneous rotation motions. These rotation motions are that tangent and normal vectors of lightlike curve rotate around each other. But, the angular speeds of them are different. Then, by doing the similar operations, we obtain that Darboux axis rotates around spacelike vector of Frenet trihedron of the lightlike curve and this spacelike vector rotates around Darboux axis. Consequently, we obtain the series of Darboux vectors by this way. So, simple mechanisms can be formed.

Walker manifolds and Killing magnetic curves

On a Walker manifold Mf3, we first characterize the Killing vector fields, aiming to obtain the corresponding Killing magnetic curves. When the manifold is endowed with a unitary spacelike vector field ξ, we prove that after a reparameterization, any lightlike curve normal to ξ is a lightlike geodesic. We also show that on Mf3, equipped with a Killing vector field V, any arc length parameterized spacelike or timelike curve, normal to V, is a magnetic trajectory associated to V. We characterize the normal magnetic curves corresponding to some Killing vector fields on Mf3, obtaining their explicit expressions for certain functions f.

Timelike rotational surfaces with lightlike profile curve

In this work, some geometric properties of the timelike rotational surfaces with lightlike profile curve of (S, L), (T,L) and (L, L) − types are shown in Minkowski 3−space.

LIGHTLIKE DEVELOPABLES IN MINKOWSKI 3-SPACE

We say that a surface in Minkowski 3-space is a lightlike developable if any pseudonormal vector of the regular part of the surface is lightlike. We show that such a surface is a part of a lightlike plane, the lightcone, the tangent surface of a spacelike curve in a lightlike plane, the tangent surface of a lightlike curve or the glue of such surfaces. The most interesting surfaces in such the class of surfaces is the tangent surface of a lightlike curve. We give a classification of the singularities for the tangent surface of a generic lightlike curve. As a consequence, the H3 type singularity appears in generic.

Conformal arc-length as $\frac 1 2$-dimensional length of the set of osculating circles

The set of osculating circles of a given curve in \boldsymbol S^3 forms a lightlike curve in the set of oriented circles in \boldsymbol S^3 . We show that its “ \frac 1 2 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

Particle with torsion on 3 d null-curves

We consider a (2 + 1)-dimensional mechanical system with the Lagrangian linear in the torsion of a light-like curve. We give Hamiltonian formulation of this system and show that its mass and spin spectra are defined by one-dimensional nonrelativistic mechanics with a cubic potential. Consequently, this system possesses the properties typical of resonance-like particles.

A self-similar curve with a light-like tangent and a fractal dimension in Minkowski space

We show that a regular curve in Minkowski space with a light-like tangent will have Hausdorff dimension D = 1/2. We present a light-like curve with D = 1/2 exhibiting complete self-similarity. We also present a Lorentz invariant formulation of the neighbourhood of a curve with "diameter" l with a volume scaling as l 4-D, where D is the fractal dimension of the curve. The connection to the theory of a massless relativistic string is briefly discussed.