Contributions to differential geometry of isotropic curves in the complex space

Abstract

This work deals with classical differential geometry of isotropic curves in the complex space C 4 . First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.