Abstract. General definition of associated curves of a Frenet curve is given in a three dimensional compact Lie group G. The principal normal direction curve and principal normal donor curve are introduced and some characterizations for these curves are obtained in G. Later, the relationships between a principal normal direction curve and some special curves such as helix, slant helix or curve with a special torsion are obtained.

In this study, we have defined slant helix according to type-2 Bishop frame in Euclidean 3-Space. Furthermore, we have given some necassary and sufficient conditons for the slant helix.

Abstract. In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space I 1 3 satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

In this paper, we study inextensible flows of curves according to type-2 Bishop frame in Euclidean 3-space. Necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature.

In this paper, we define tubular surface by using a Darboux frame instead of a Frenet frame. Subsequently, we compute the Gaussian curvature and the mean curvature of the tubular surface with a Darboux frame. Moreover, we obtain some characterizations for special curves on this tubular surface in a Galilean 3-space.

In this paper, we study tube surfaces with type-2 Bishop frame instead of Frenet frame in Euclidean 3-space E 3 .Besides, we have discussed Weingarten and linear Weingarten conditions for tube surfaces with the Gaussian curvature K, the mean curvature H and the second Gaussian curvature K II .

In this study, we define a new type of associated curves in the Euclidean 3-space such as normal-direction curve and normal-donor curve. We obtain characterizations for these curves. Moreover, we give applications of normal-direction curves to some special curves such as helix, slant helix, plane curve or normal-direction (ND)-normal curves in E 3 . And, we show that slant helices and rectifying curves can be constructed by using normal-direction curves.

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