The Darboux Trihedrons of Regular Curves on a Regular Time-Like Surface

Emin Özyilmaz

doi:10.3390/mca19020172

Abstract

In this work, we study Differential Geometry in the Minkowski 3-space R13 of the curves on a time-like regular surface by using parameter curves which are not perpendicular to each other. The aim of this study is to investigate the formulae between the Darboux Vectors of the time-like curve (c) , the time-like parameter curve (c1) and the space-like parameter curve (c2) which are not intersecting perpendicularly.

Highlights

Classical differential geometry of the curves may be surrounded by the topics which are general helices, involute-evolute curve couples, spherical curves and Bertrand curves
At the beginning of the twentieth century, Einstein’s theory opened a door to new geometries such as Minkowski space-time, which is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold
The theory of degenerate submanifolds has been treated by researchers and some of the classical differential geometry topics have been extended to Lorentzian manifolds

Summary

INTRODUCTION

Classical differential geometry of the curves may be surrounded by the topics which are general helices, involute-evolute curve couples, spherical curves and Bertrand curves. Such special curves are investigated and used in some of real world problems like mechanical design or robotics by well-known Frenet-Serret equations. In the light of the available literature, in [4] the author extended spherical images of curves to a four-dimensional Lorentzian space and studied such curves in the case where the base curve is a space-like curve according to the signature (+,+,+,-). We investigate the formulae between the Darboux Vectors of the curve (c) , the parameter curves (c1) and (c2 ) which are not intersecting perpendicularly. We will find an opportunity to investigate regular time-like surface by taking the parameter curves which are intersecting under the angle

PRELIMINARIES

THE DARBOUX VECTOR FOR THE DARBOUX TRIHEDRON OF A TIME-LIKE CURVE

Tg ch Rn

Result