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

Fatma Bulut

doi:10.3390/sym13112185

Abstract

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.

Highlights

IntroductionCurve theory is the most important area of work
In classical differential geometry, curve theory is the most important area of work.Special curves and their characterizations have been studied for a long time and are still being studied
In three-dimensional Euclidean space, the Darboux frame is the velocity of the curve and is formed by the normal vector of the surface, whereas the Frenet–Serret frame is created from the acceleration and velocity of the curve

Summary

Introduction

Curve theory is the most important area of work Special curves and their characterizations have been studied for a long time and are still being studied. In three-dimensional Euclidean space, the Darboux frame is the velocity of the curve and is formed by the normal vector of the surface, whereas the Frenet–Serret frame is created from the acceleration and velocity of the curve. Spacelike normal curves in E41 whose Frenet frame contains only non-null vector fields, as well as the timelike normal curves in E41 , in terms of their curvature functions and some special spacelike curves in Minkowski space-time E41 were constructed, respectively, by [2] and [3]. An extended Darboux frame field along a nonnull curve lying on an orientable non-null hypersurface in Minkowski space-time was presented by Duldul [4].

Geometric Preliminaries

Differential Geometry of the ED-Frame in Minkowski 4-Space

Conclusions