Differential geometry of special type curves

In this thesis, it is aimed to examine AW(k)-type curves in 3-dimensional Euclid and 3-dimensional Lorentz space. The close relationship between AW(k)-type curves and Bertrand curves and helices has been examined. AW(k)-type curves are a space curves that contain relations between the first and second curvatures of the curve. Our study consist of 5 parts. In the Introduction and Basic Consepts chapters, which are the first and second chapters, curves, AW(k)-type curves, basic knowledge of curves are mentioned. In the third chapter, the fundamental definitions and theorems that we will utilize in our calculations in 3-dimensional Euclid space and in 3-dimensional Lorentz space are given, Frenet frame, helix, slant helix, conical geodesic curve, Bertrand curve and AW(k)-type curve are introduced. In the fourth chapter, AW(k)-type curves in 3-dimensional Euclid space, weak AW(k)-type curves, the relations of these curves with Bertrand curve and conical geodesic curve are examined and theorems related to them have been proved. In the fifth chapter, harmonic curvatures of a Frenet curve are examined. Moreover, AW(k)-type curves in 3-dimensional Lorentz space are expressed and proved. In Lorentz space, the relations of AW(k)-type curves with harmonic curvatures are given.