Null Curves Research Articles

Integrable flows on null curves in the Anti-de Sitter 3-space

Abstract We formulate integrable flows related to the Korteweg–De Vries (KdV) hierarchy on null curves in the anti-de Sitter 3-space ( AdS ). Exploiting the specific properties of the geometry of AdS , we analyze their interrelationships with Pinkall flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lamé equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

Relation between the minimal Lorentz surfaces in ℝ24 and ℝ13

In this paper, we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space [Formula: see text] using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in [Formula: see text] and the pairs of null curves in [Formula: see text]. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in [Formula: see text] and the pairs of minimal Lorentz surfaces in [Formula: see text]. As an application, we give examples showing how we can obtain a minimal Lorentz surface in [Formula: see text] using a pair of minimal Lorentz surfaces in [Formula: see text].

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We prove that on every compact Riemann surface M, there is a Cantor set C⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\subset M$$\\end{document} such that M\\C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M{ \\setminus }C$$\\end{document} admits a proper conformal constant mean curvature one (CMC-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {CMC-1}$$\\end{document}) immersion into hyperbolic 3-space H3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^3$$\\end{document}. Moreover, we obtain that every bordered Riemann surface admits an almost proper CMC-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {CMC-1}$$\\end{document} face into de Sitter 3-space S13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}_1^3$$\\end{document}, and we show that on every compact Riemann surface M, there is a Cantor set C⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\subset M$$\\end{document} such that M\\C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M {\\setminus } C$$\\end{document} admits an almost proper CMC-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {CMC-1}$$\\end{document} face into S13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}_1^3$$\\end{document}. These results follow from different uniform approximation theorems for holomorphic null curves in C2×C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^2 \ imes \\mathbb {C}^*$$\\end{document} that we also establish in this paper.

On null Cartan normal isophotic and normal silhouette curves on a timelike surface in Minkowski 3‐space

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature and constant geodesic torsion . We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non‐zero constant curvatures and have a remarkable property that they are general helices, relatively normal‐slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions.

Twisted Surfaces in Semi-Euclidean $4$-Space with Index $2$

In this paper, we consider the twisted surfaces in semi-Euclidean 4-space with index 2.We classify the twisted surface with respect to their spine curve which are non-null or null curves. So, we study the geometric properties of these surfaces. Also we obtain the family of some special surfaces such as flat surfaces, marginally trapped surfaces.

On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$

In this paper, we introduce generalized Darboux frames of the first and the second kind along a null Cartan curve lying on a timelike surface in Minkowski space ${E}^{3}_{1}$ and define null Cartan rectifying isophotic and rectifying silhouette curves in terms of the vector field that belongs to generalized Darboux frame of the first kind. We investigate null Cartan rectifying isophotic and rectifying silhouette curves with constant geodesic curvature $k_g$ and geodesic torsion $\tau_g$ and obtain the parameter equations of their axes. We prove that such curves are the null Cartan helices and the null Cartan cubics. We show that the introduced curves with a non-zero constant curvatures $k_g$ and $\tau_g$ are general helices, relatively normal-slant helices and isophotic curves with respect to the same axis. In particular, we find that null Cartan cubic lying on a timelike surface is rectifying isophotic and rectifying silhouette curve having a spacelike and a lightlike axis. Finally, we give some examples.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

Curves whose curvature depends on their position and null curves

We show that, apart from degeneracies, determining a plane curve whose curvature depends on its position essentially consists of determining a null curve in the Lorentzian 3-space when the null tangent direction depends on its position. We use this point of view to investigate the intrinsic equations for the n-elastic curves. We show how the problem of prescribed null tangent direction in terms of the position can be solved by quadratures when the prescription exhibits sufficient symmetries. This problem is formalized in terms of a convenient contact 3-form.

Null cartan geodesic isophote curves in Minkowski 3-space

In this paper, we investigate null Cartan geodesic isophote curves in the Minkowski [Formula: see text]-space, and give examples where such curves actually exist. By categorizing the types of light vectors, we characterize different types of null Cartan geodesic isophote curves. Moreover, we present the relationship between null Cartan geodesic isophote curves and other special curves.

The geometrical properties of the Smarandache curves on 3-dimension pseudo-spheres generated by null curves

<p>In this paper, we presented the geometrical properties of Smarandache curves on 3-pseudo-spheres. These curves were examined in the context of the lightcone, de Sitter space, and anti-de Sitter space. By leveraging the curvature relationships between the null curve and its corresponding Smarandache curves, we established necessary and sufficient conditions. Additionally, we illustrated our main results through two examples.</p>

BMS3 (Carrollian) field theories from a bound in the coupling of current-current deformations of CFT2

Two types of Carrollian field theories are shown to emerge from finite current-current deformations of toroidal CFT2’s when the deformation coupling is precisely fixed, up to a sign. In both cases the energy and momentum densities fulfill the BMS3 algebra. Applying these results to the bosonic string, one finds that the electric-like deformation (positive coupling) reduces to the standard tensionless string. The magnetic-like deformation (negative coupling) yields to a new theory, still being relativistic, devoid of tension and endowed with an “inner Carrollian structure”. Classical solutions describe a sort of “self-interacting null particle” moving along generic null curves of the original background metric, not necessarily geodesics. This magnetic-like theory is also shown to be recovered from inequivalent limits in the tension of the bosonic string. Electric- and magnetic-like deformations of toroidal CFT2’s can be seen to correspond to limiting cases of continuous exactly marginal (trivial) deformations spanned by an SO(1,1) automorphism of the current algebra. Thus, the absolute value of the current-current deformation coupling is shown to be bounded. When the bound saturates, the deformation ceases to be exactly marginal, but still retains the full conformal symmetry in two alternative ultrarelativistic regimes.

A novel approach to the motion equations of null Cartan curves via the compatible Hasimoto map

In this manuscript, we try to understand the links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of null Cartan curves and some physical results. First, we obtain the equations of Hasimoto transformation occurred by applying Hasimoto map for null Cartan curves. Then, based on the Hasimoto map of null Cartan curves, we examine both Landau–Lifshitz system and visco Landau–Lifshitz system. Lastly, we portray the links of findings obtained by geometric physical issues. After all, we propose a new method, known as Khater method, to obtain solutions of fractional order differential equations. Also, these solutions are simulated by graphically using physical constructions.

Involutes of null Cartan curves and their representations in Minkowski 3-space

Involutes of null Cartan curves and their representations in Minkowski 3-space

{L_1}-2-Type Surfaces in 3-Dimensional De Sitter and Anti De Sitter Spaces

Let M_s^2 be an orientable surface immersed in the De Sitter space mathbb {S}_1^3subset mathbb {R}^4_1 or anti de Sitter space mathbb {H}_1^3subset mathbb {R}^4_2. In the case that M_s^2 is of L_1-2-type we prove that the following conditions are equivalent to each other: M_s^2 has a constant principal curvature; M_s^2 has constant mean curvature; M_s^2 has constant second mean curvature. As a consequence, we also show that an L_1-2-type surface is either an open portion of a standard pseudo-Riemannian product, or a B-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant.

Generalized Hasimoto-type surfaces of null growth in Minkowski 3-space

In this article, the idea of the generalized Hasimoto-type surfaces are put forward based on the interaction between vortex filaments. Meanwhie, the surface of null growth is proposed by evolving a null curve as dictated direction and growth velocity in Minkowski 3-space. The conditions and geometric forms of the generalized Hasimoto-type surfaces of null growth are investigated. Last but not least, several typical examples are presented to characterize such surface growth and the corresponding perturbations explicitly.

New results concerning Cartan null and pseudo null curves in Minkowski 3-space

New results concerning Cartan null and pseudo null curves in Minkowski 3-space

Almost Robinson geometries

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy–Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.

On Generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space

In this paper we define generalized Darboux frame of a a pseudo null curve $\alpha$ lying on a lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that $\alpha$ has two such frames and obtain generalized Darboux frame's equations. We obtain the relations between the curvature functions of $\alpha$ with respect to the Darboux frame and generalized Darboux frames. We also find parameter equations of the Darboux vectors of the Frenet, Darboux and generalized Darboux frames and give the necessary and the sufficient conditions for such vectors to have the same directions. Finally, we present related examples.

On Killing Magnetic Fields of Magnetic Pseudo Null Curves in R_1^3

In present paper, we study the properties of the Killing magnetic field of a pseudo null magnetic curve in R_1^3 . We investigate for the T, N and B-magnetic cases of a pseudo null curve. We find the conditions for the Killing magnetic field to be pseudo null and derive the Frenet formulas for this field. Finally, we obtain the conditions for the transformation of the pseudo null curve to its pseudo null Killing magnetic field with equal torsions.

Inextensible Flows of Null Cartan Curves in Minkowski Space R2,1

This research focused on studying the flows of a null Cartan curve specified by the velocity and acceleration fields. We have proven that the tangential and normal velocities are influenced by the binormal velocity along the motion. The velocity fields are used to drive the time evolution equations for the Cartan frame and the torsion of the null curve. The objective of this work is to construct a family of inextensible null Cartan curves from the flows of the initial null Cartan curve. The surface formed by this family of inextensible flows of the null Cartan curve is obtained numerically and visualized. In this paper, we refer to the surface traced out by the family of the null Cartan curve as the generated or constructed surface. We present a novel model for the inextensible null Cartan curve, which moves with a constant binormal velocity to describe the process of constructing a family of null Cartan curves. Through this model, the time evolution equation for the torsion of the inextensible null Cartan curve arises as the Korteweg-de Vries (K-dV) equation, and we obtain and visualize the soliton solutions. The soliton solutions represent the torsion of the family of null Cartan curves at various time values. We construct the family of inextensible null Cartan curves and visualize the generated surface. In addition, we investigate the flows of inextensible null Cartan curves specified by acceleration fields, and we obtain the explicit relationships between the acceleration and velocity functions. Finally, we provide an application for the inextensible flows of the null Cartan curve with constant normal acceleration.