Classical Differential Geometry of Curves According to Type-2 Bishop Trihedra

Emin Özyılmaz

doi:10.3390/mca16040858

Abstract

In this work, we study classical differential geometry of the curves according to type-2 Bishop trihedra. First, we present some characterizations of a general helix, a helix, special cases and spherical curves. Thereafter, we investigate position vector of a regular curve by a system of ordinary differential equations whose solution gives the components of the position vector with respect to type-2 Bishop frame. Next we prove that the first vector field of the type-2 Bishop frame of a regular curve satisfies a vector differential equation of third order. Solutions of the mentioned system and vector differential equation have not been found. Therefore we present some special characterizations introducing special planes of three dimensional Euclidean space.

Highlights

Classical differential geometry of the curves may be surrounded by the topics general helices, involute-evolute curve couples, spherical curves and Bertrand curves
We prove that the first vector field of the type-2 Bishop frame of a regular curve satisfies a vector differential equation of third order
Spherical images, the tangent and the binormal indicatrix and some characterizations of such curves in Euclidean space and Lorentz-Minkowski spaces are presented. This fact has been extended to the isotropic curves in the complex space by [23]. At this point one can ask that the first vector field of the type-2 Bishop frame ζ1 forms a constant angle with a fixed a direction or not? This is an interesting question and we investigate this case with the following statements

Summary

I TRODUCTIO

Classical differential geometry of the curves may be surrounded by the topics general helices, involute-evolute curve couples, spherical curves and Bertrand curves. Bishop pioneered “Bishop Frame” by means of parallel vector fields This special frame is called as “parallel” or “alternative” frame of the curves. In [21], the authors introduced a new version of the Bishop frame and called it as “type-2 Bishop frame” They investigated spherical images of a regular curve which correspond to each vector fields of the new trihedra. We prove that the first vector field of the type-2 Bishop frame of a regular curve satisfies a vector differential equation of third order. We hope these results will be helpful to mathematicians who are specialized on mathematical

PRELIMI ARIES

SOME CHARACTERIZATIO OF REGULAR CURVES ACCORDI G to TYPE-2 BISHOP FRAME

POSITIO VECTOR OF A REGULAR CURVE ACCORDĐ G TO TYPE-2

VECTOR DIFFERE TIAL EQUATIO OF THIRD ORDER CHARACTERIZES REGULAR CURVES OF E3