scholarly journals Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2185
Author(s):  
Fatma Bulut
Keyword(s):  
Differential Geometry ◽  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Curve ◽  
Frame Field ◽  
Dimensional Minkowski Space ◽  
Darboux Frame ◽  
Surface Curve ◽  
Area Of Study

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

scholarly journals N-BISHOP DARBOUX VECTOR OF A TIMELIKE CURVE IN MINKOWSKI 3-SPACE

2020 ◽  
Vol 20 (3) ◽  
pp. 519-528
Author(s):  
HATICE KUSAK SAMANCI
Keyword(s):  
Differential Geometry ◽  
Minkowski Space ◽  
Timelike Curve ◽  
Rotation Vector ◽  
Effective Alternative ◽  
Darboux Vector ◽  
Bishop Frame ◽  
Alternative Approach ◽  
Darboux Frame ◽  
First Time

It is known that a Bishop frame of a curve is one of the effective alternative approach in the differential geometry. Recently, several important works have been done about the Bishop frames. The aim of our paper is to investigate the N-Bishop frame for timelike curves in Minkowski space. We define the N-Bishop frame for the timelike curve in Minkowski space. Then, we consider some properties of the frame. Moreover, we describe the N-Bishop Darboux frame for the first time. Additionally, we compute some geometrical characterizations for the N-Bishop Darboux axis and momentum rotation vector.

scholarly journals ON NONLINEARITY OF p-BRANE DYNAMICS

2012 ◽  
Vol 09 (06) ◽  
pp. 1261017 ◽  
Author(s):  
A. A. ZHELTUKHIN
Keyword(s):  
Exact Solutions ◽  
Minkowski Space ◽  
Nonlinear Equations ◽  
New Exact Solutions ◽  
Dimensional Minkowski Space

Nonlinear equations of p-branes in D = (2p + 1)-dimensional Minkowski space are discussed. Presented are new exact solutions for a set of spinning p-branes with the Abelian symmetries U(1) × U(1) × ⋯ ×U(1) of their shapes.

scholarly journals New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

Geometry ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
İsmail Aydemir ◽  
Fırat Yerlikaya
Keyword(s):  
Minkowski Space ◽  
3 Dimensional ◽  
Bertrand Curve ◽  
Dimensional Minkowski Space ◽  
Characteristic Features

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection

1990 ◽  
Author(s):  
Joseph Pegna ◽  
Franz-Erich Wolter
Keyword(s):  
Differential Equation ◽  
Orthogonal Projection ◽  
Broad Class ◽  
Design Tool ◽  
Space Curve ◽  
Computationally Efficient ◽  
Curves On Surfaces ◽  
Surface Representations ◽  
Differential Properties ◽  
Surface Curve

Abstract In the design and manufacturing of shell structures it is frequently necessary to construct trimming curves on surfaces. The novel method introduced in this paper was formulated to be coordinate independent and computationally efficient for a very general class of surfaces. Generality of the formulation is attained by solving a tensorial differential equation that is formulated in terms of local differential properties of the surface. In the method proposed here, a space curve is mapped onto the surface by tracing a surface curve whose points are connected to the space curve via surface normals. This surface curve is called to be an orthogonal projection of the space curve onto the surface. Tracing of the orthogonal projection is achieved by solving the aforementionned tensorial differential equation. For an implicitely represented surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This method has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations. Orthogonal projection of a curve onto a surface was also developed in the context of surface blending. The orthogonal projection of a curve onto two surfaces to be blended provides not only a trimming curve design tool, but it was also used to construct smooth natural maps between trimming curves on different surfaces. This provides a coordinate and representation independent tool for constructing blend surfaces.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

The two-dimensional minkowski space fermion determinant by example

1988 ◽  
Vol 211 (1-2) ◽  
pp. 107-110 ◽  
Author(s):  
D. Cangemi ◽  
M. Makowka ◽  
G. Wanders
Keyword(s):  
Minkowski Space ◽  
Two Dimensional ◽  
Dimensional Minkowski Space
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