Null Helices Research Articles

Constant pseudo-angle lightlike surfaces

The oriented angles between lightlike vectors cannot be defined properly compared to the timelike vectors in the Minkowski spacetime. Therefore, we use the pseudo-angles between any non-lightlike or lightlike vectors to develop the theory of lightlike surfaces having constant angle with a fixed nonlightlike direction. We investigate some geometric properties on these surfaces such as being a tangent developable. Besides, we construct the constant angle lightlike ruled surfaces by means of the null helices. We give several examples to illustrate the obtained surfaces.

Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results.

Darboux Associated Curves of a Null Curve on Pseudo-Riemannian Space Forms

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained.

Structure Functions of Pseudo Null Curves in Minkowski 3-Space

In this work, the embankment surfaces with pseudo null base curves are investigated in Minkowski 3-space. The representation formula of pseudo null curves is obtained via the defined structure functions and the k-type pseudo null helices are discussed completely. Based on the theories of pseudo null curves, a class of embankment surfaces are constructed and characterized by the structure functions of the pseudo null base curves.

Null helices and Cartan slant helices in Lorentz–Minkowski 3-space

In this paper, we study null helices, Cartan slant helices and two special developable surfaces associated to them in Lorentz–Minkowski 3-space. We give a method using a special plane curve to construct a null helix. We also define the null tangential Darboux developable of a null Cartan curve, and we give a classification of singularities of it. Moreover, we study the relationship between null helices (or Cartan slant helices) with the developable surfaces of them.

Some Characterizations of Generalized Null Scrolls

In this work, a family of ruled surfaces named generalized null scrolls in Minkowski 3-space are investigated via the defined structure functions. The relations between the base curve and the ruling flow of the generalized null scroll are revealed. The Gaussian curvature, mean curvature, second Gaussian curvature and the second mean curvature are given and related to each other. Last but not least, the generalized null scrolls whose base curves are k-type null helices are discussed and several examples are presented.

Conformal theory of curves with tractors

We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a specification of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classical Frenet frame construction and then expressed in terms of relative ones. Our approach applies likewise to conformal structures of any signature; in the case of indefinite signature we focus especially on the null curves. It also provides a conceptual tool for handling distinguished families of curves (conformal circles and conformal null helices) and conserved quantities along them.

Null Quaternionic Bertrand Curves in Minkowski Space R14

The geometry of null curves in Minkowski spacetime has played an important role in the development of general relativity, as well as in mathematics and physics of gravitation. Many scientists have used Minkowski space to apply general relativity. There has been an increase in research on null curves in geometry and physics [1]. Bonnor [2] has introduced a Cartan frame for null curves in R 1 and proved the fundamental existence and congruence theorems. Bejancu [3] has given a method for the general study of the geometry of null curves in lightlike manifolds and in semi-Riemannian manifolds. A. Ferrandez, A. Gimenez and P. Lucas [4] have shown that a null Frenet curve, parametrized by the pseudo-arc parameter, is a null helix, if its lightlike curvature is constant. Coken and Ciftci [5] have reconstructed the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and have characterized the Bertrand null curves. And then, Duggal and Jin [6] have studied major developments of null curves, hypersurfaces and their physical use, in their recent book with voluminous bibliography. Quaternions were discovered by Hamilton as an extension to the complex numbers in 1843. Quaternions have found a broad application in many scientific areas: in mechanics of a solid body, for the description of rotation in space, in computer animation, etc. One of the most important tools used to analyze a quaternionic curve is the Frenet frame. Therefore, in [7], Bharathi and Nagaraj have defined Serret-Frenet formulas for a quaternionic curve in E and E, and then Coken and Tuna have studied Serret-Frenet formulas for quaternionic curves and quaternionic inclined curves in semi-euclidean spaces [8]. Moreover, we have studied the differential geometry of null quaternionic curves in semi-euclidean 3-spaces R v and gave the Frenet formula for null quaternionic curves by using spatial quaternions. We have constructed recently the Cartan frame for a null quaternionic curve in the 4-dimensional Minkowski space R 1 [9]. Then, we have established a relation of Bertrand pairs with null quaternionic Cartan helices in R v [10].

On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

General Helices with Timelike Slope Axis in Minkowski 3-Space

In the present paper, we consider the general helices in Minkowski 3-space to have a constant timelike slope axis. As a result, we show that there exists no pseudo null helix with constant timelike slope axis or spacelike helix with timelike principal normal and constant timelike slope axis. Moreover, we obtain the parametric equations of spacelike helix with spacelike principal normal, timelike helix and null Cartan helix with timelike slope axis. We also give some related examples and their figures.

Generalized Null Bertrand Curves In Minkowski Space-Time

Abstract Çöken and ÇIFTCI proved that a null Cartan curve in Minkowski space-time E41 is a null Bertrand curve if and only if k2 is nonzero constant and k3 is zero. That is, the null curve with non-zero curvature k2 is not a Bertrand curve in Minkowski space-time E41. So, in this paper we defined a new type of Bertrand curve in Minkowski space-time E41 for a null curve with non-zero curvature k3 by using the similar idea of generalized Bertrand curve given by Matsuda and Yorozu and we called it a null (1, 3)-Bertrand curve. Also, we proved that if a null curve with non-zero curvatures in Minkowski space-time E41 is a null (1, 3)-Bertrand curve then it is a null helix. We give an example of such curves.

BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS

Let M 3(c) denote the 3-dimensional space form of index q = 0,1, and constant curvature c 6 0. A curveimmersed in M 3(c) is said to be a Bertrand curve if there exists another curveand a one-to-one correspondence betweenandsuch that both curves have common principal normal geodesics at corresponding points. We obtain charac- terizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non- null Bertrand curves in M3(c) correspond with curves for which there exist two constants � 6 0 and µ such that �� + µ� = 1, whereand � stand for the curvature and torsion of the curve. As a consequence, non-null helices in M 3(c) are the only twisted curves in M 3(c) having in- finite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

NOTE ON NULL HELICES IN <TEX>$\mathbb{E}_1^3$</TEX>

In this paper, we study null helices, null slant helices and Cartan slant helices in . Using some associated curves, we characterize the null helices and the Cartan slant helices and construct them. Also, we study a space-like curve with the principal normal vector field which is a degenerate plane curve.

Null Darboux developable and pseudo-spherical Darboux image of null Cartan curve in Minkowski 3-space

Singularities of null Darboux developable, Gaussian surfaces and pseudo-spherical Darboux images associated with a null Cartan curve will be investigated in Minkowski 3-space. The relationships will be revealed between singularities of the above three subjects and differential geometric invariants of null Cartan curves, these invariants are deeply related to the order of contact of null Cartan curves with null helices.

On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space

We investigate null curves of the AW(k)-type (1 ≤ k ≤ 3) in the 3-dimensional Lorentzian space, L3, and give curvature conditions of these curves by using the Cartan frame. Moreover, we study harmonic curvatures of curves of AW(k)-type and show that if the α Frenet curve is of type AW(1), then α is a null helix.

Position vectors of a timelike and a null helix in Minkowski 3-space

In this paper, we study the position vectors of a timelike and a null helix (or a W-curve), i.e. curve with constant curvatures in the Minkowski 3-space E 1 3 . We give some characterizations for timelike and null helix whose image lies on the Lorentzian sphere S 1 2 or pseudo-hyperbolical space H 0 2 by using the positions vectors of the curve.

Relativistic particles and the geometry of 4-D null curves

We study actions in ( d + 1 ) -dimensions associated with null curves, mainly when d = 3 , whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler–Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.

Harmonic curvatures of null curves and the null helix in R_1^{m+2}

Harmonic curvatures of null curves and the null helix in R_1^{m+2}

Null Generalized Helices in L3 and L4, 3 and 4-Dimensional Lorentzian Space

In this paper we obtain the harmonic curvatures of a null generalized helix in L3 and L4 , 3 and 4-dimensional Lorentzian spaces. Thus we another definitions of null generalized helix in terms of harmonic curvatures. Also we find the relation of between Frenet frame, slope axis and curvature functions 1 k , 2 k of null generalized helix. Finally we characterize the null helices thanks to harmonic curvatures in L3 and L4 , 3 and 4- dimensional Lorentzian space.

Null generalized helices in Lorentz Minkowski spaces

We obtain a Lancret-type theorem for null generalized helices in Lorentz– Minkowski spaces L n .I nL 3 we find that the only null generalized helices are the ordinary null helices. However, in L 5 we have to consider two types of null generalized helices according to whether the axis is non-null or null. In both cases we obtain the solutions to the natural equations problem.