Characterizations of Space Curves With 1-Type Darboux Instantaneous Rotation Vector

Abstract

Abstract. In this study, by using Laplace and normal Laplace operators, we give some characterizations for the Darboux instantaneous rotation vector field of the curves in the Euclidean 3-space E 3 . Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vectors. Moreover, we obtain some characterizations of helices according to Darboux vector.

Some New Characterizations of The Harmonic and Harmonic 1-Type Curves in Euclidean 3-Space

“…curve , respectively. Also, the operator is called by the normal Laplace-Beltrami operator with respect to [9][10][11]2].…”

“…(2013) obtained some characterizations of the timelike curves using the Darboux vector in Minkowski 3-space [19]. Arslan et al made a similar study using a 1-type Darboux instantaneous rotation vector [2]. Subsequently, Kocayiğit and his colleagues (2016) calculated some differential equation characterizations of space curves due to Bishop frame in Euclidean 3-space [20].…”

Harmonicity and differential equation of involute of a curve in E3

“…-If ΔH = λH then α is called a 1-type harmonic curve, λ ∈ R, -If ΔH = 0 then α is called a biharmonic curve [7,8].…”

“…Thanks to meticulous studies, it has been revealed that curves can be classified [6]. After this classification, a great many number of articles have been written, [7,8] and also [9]. In this paper, we first take a unit speed curve which we call through the work as main curve, then write the characterizations of an involute curve by means of Frenet apparatus of the main curve.…”

“…is called a Laplace operator [7,8]. Let α be the unit speed curve and H be the mean curvature vector field along the curve α.…”