Related papers: $(1,2)$-Null Bertrand Curves in Minkowski Spacetim…
We investigated the evolute of a space curve with singular points. As smooth curves with singular points, we apply the theory of framed curves. However, the involute corresponding to the evolute in the sense of the locus of the centre of…
We give a definition of generalized timelike Mannheim curve in Minkowski space-time $E_1^4$. The necessary and sufficient conditions for the generalized timelike Mannheim curve obtain. We show some characterizations of generalized Mannheim…
In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike…
Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth…
The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves…
Assume that arc length is measured with the flat spacetime metric. Then, for the most general Poincare group representation for translating 4-vectors, curves with parallel translated tangent vectors must have accelerations that are the…
In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean…
We study the moduli space of null curves in Klein's quartic in the four-dimensional (complex) projective plane using methods developed by Robert Bryant. As a consequence, we show that minimal surfaces with $9$ embedded planar ends do not…
We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we…
In this work, first, we express some characterizations of helices and ccr curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret invariants of Bertrand curve of a helix are presented. Moreover, in the same space, some…
In this paper, new representations of a Bertrand curve pair in three dimensional Lie groups with bi-invariant metric are given. Besides, the spherical indicatrices of a Bertrand curve pair are obtain and the relations between the spherical…
Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves…
We introduced generalized Bishop frames on curves in 4-dimensional Euclidean space $\mathbb{E}^{4}$, which are orthonormal frames such that the derivatives of the vectors of the frames along the curve can be expressed, via a certain matrix,…
We construct a class of perturbations of the Cauchy-Riemann equations for maps from curves to a Calabi-Yau threefold. Our perturbations vanish on components of zero symplectic area. For generic 1-parameter families of perturbations, the…
Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a…
We analyse the existence of closed timelike curves in spacetimes which possess an isometry. In particular we check which discrete quotients of such spaces lead to closed timelike curves. As a by-product of our analysis, we prove that the…
In the present paper, we give new Frenet formulas for the Bertrand partner curve by taking the advantage of relations between curvatures and a curve itself. Then making use of these formulas we write the differential equations and…
We consider the region of closed timelike curves (CTC's) in three-dimensional flat Lorentz spacetimes. The interest in this global geometrical feature goes beyond the purely mathematical. Such spacetimes may be considered lower-dimensional…
In the theory of differential geometry curves, a curve is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. In this paper, we study and characterize…
In this paper, we study inextensible flows of non-null curves in E^n,1. We give necessary and sufficient conditions for inextensible flow of nonnull curves in E^n,1.