Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M , have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, b...

We have derived a necessary and sufficient condition for non-null normal spacelike curve lying in or timelike surface M ⊂ E13, so that the becomes K-type slant helix with K ∈ {1,2,3}. used Darboux frame to define conditions. An example is given 1-type having binormal.

The aim of the paper is on spacelike circular surfaces and singularities in Minkowski 3-space E13. A surface with a stationary radius can be swept out by movable Lorentzian circle following non-null curve, which acts as spine curve. In study, we have represented furnished its geometric properties such striction curves contrasting those ruled surfaces. Subsequently, new type was distinguished na...

A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder, cone) is a constant angle surface if and only if the generating curve is a ...

By using the Darboux frame |?, ?, ?| of a non-null curve lying on timelike surface in Minkowski 3-space, where ? is unit tangent vector curve, spacelike normal field restricted to and = ?? we define relatively normal-slant helices as curves satisfying condition that scalar product fixed spanning their axis non-constant constant. We give necessary sufficient conditions for be helices. consider s...

In this work, we firstly introduce notions of principal directed curves and donor which are associated a Frenet curve in the dual Lorentzian space D 3 1 D13 . We give some relations between curvature torsion curve. show that general helix is plane obtain equation by using position vector Then circle $\mathbb{D}^{2}$ or hyperbola $\mathbb{D}_{1}^{2}$ slant $\mathbb{D}_{1}^{3}$ helix, respectivel...

In this study, the parametric equation of hypersurfaces passing through spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with help curve is frame. Furthermore, hypersurface families were created by giving necessary sufficient conditions so on common isoasymptotic hypersurface. Then, these obtained deviation marching-scale functions si...

Abstract In this paper, we study the vortex filament flow for timelike and spacelike curves in Minkowski 3-space. The equations of are equivalent to nonlinear Schrödinger equation heat equation, respectively. As a consequentce, prove that soliton curve gives solution traveling wave on line at infinity. Also, terms new complex frame. Finally, discuss method find exact shape from by solving Frene...

In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along prescribed geometric flows (including classical curve shortening flow or mean curvature as a special case), which correspond to class quasilinear parabolic initial boundary value problems, and can prove that exists for all time. Moreover, also show evolving converge straight lin...