Differential geometry of curves with fractional derivative

The theory of curves has been an interesting field of study in differential geometry for a long time, but recently using fractional derivative techniques in this theory has brought a new perspective to many studies. The aim of this doctoral thesis is to examine the basic properties of curves in differential geometry with the help of the Caputo fractional derivative. Within the scope of this doctoral thesis, firstly, the arc length of plane curves in Euclidean space, Frenet-Serret framework and Frenet-Serret formulas were examined and the fractional curvatures of this plane curve were given in detail. Then, the fundamental theorems of fractional space curves in 3-dimensional Euclidean space and the Frenet-Serret formulas are discussed. The results given in this section are compared with the results obtained in the classical sense. In the last chapter, fractional invariants of curves in high-dimensional Euclidean spaces are obtained and the relations between the curvatures of some special curves are given. The suggestions regarding the method used throughout the thesis, the contribution of the obtained results to the literature and the problems are given in the last part.