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 rotationvector field of the curves in the Euclidean 3-space E 3 . Further, we givenecessary and sufficient conditions for unit speed space curves to have1-type Darboux vectors. Moreover, we obtain some characterizations ofhelices according to Darboux vector. 1. IntroductionOne of the most important problems of local differential geometry is toobtain the relations characterizingspecial curves with respect to their curvatureand torsion. The well-known types of such special curves are constant slopecurves or general helices which are defined by the property that the tangentvectors of curves make a constant angle with fixed directions. A necessaryand sufficient condition for a curve to be a general helix in the Euclidean 3-space E 3 is that the ratio of curvature to torsion is constant [11]. So, manymathematicians have focused their studies on these special curves in differentspaces such as Euclidean space and Minkowski space [3, 4, 5, 10].Furthermore, Chen and Ishikawa [1] classified biharmonic curves, the curvesfor which ∆H~ = 0 holds in semi-Euclidean space E