Applications of fractional calculus to differential geometry

For the last decade, using the techinques of fractional calculus in the differential geometric theories has been an interesting field. The aim of this master thesis is to analysis the differential geometric surfaces via Caputo fractional derivative. As a first step, in the context of this thesis, the notion of local surface of fractional order is introduced. Following this notion, the local surfaces of fractional order are characterized by their second fundamental form of fractional order. Clearly, it is obtained that a local surface of fractional order is geometrically a part of plane when the components of second fundamental form of fractional order vanish identically. This coincides with the classical situation. Then we provide that the geometrical quantities of fractional are invariant under the Euclidean rigid motions. The cylindrical surfaces representing an important class of differential geometric surfaces are considered. A parametrization of those surfaces of fractional order is defined, obtaining that those surfaces are flat of fractional order. In addition, we obtain that a cylindrical surface is a part of plane if it is minimal of fractional order. This coincides with the classical situation again. The ideas of this thesis are supported by examples with figures. In the last section, we discuss pros and cons of the used method and obtained results as well as the proposed open problems