Generalized Fermi-Walker derivative and geometric applications

In this doctoral thesis, the introduction of the subject has been given in the first section. Basic definitions and concepts, which are necessary for the thesis have been given in the second section. Then the definitions of Fermi-Walker derivative and generalized Fermi-Walker derivative have been given. By using the concepts of Fermi-Walker parallelism and non-rotating frame, generalized Fermi-Walker parallelism and generalized non-rotating frame concepts have been defined. In the third section, materials and methods which are used for preparation of this doctoral thesis have been given. The fourth section has been divided into three parts. In the first part, Fermi-Walker derivative has investigated in Euclidean 3-space. Initially the conditions of the generalized Fermi-Walker parallelism of any vector field along any curve are analyzed in Euclidean 3-space. It has shown that Frenet frame is generalized non-rotating frame along all curves. Similar operations have been made for Darboux frame considering any curve on any surface. After that, the conditions of being generalized non-rotating frame has been investigated for Bishop frame. In the second part, generalized Fermi derivative has been defined along any curve in the hypersurfaces by using the concept of Fermi derivative. The investigation of the conditions of generalized Fermi parallelism of any vector field which is along a curve on any surface in Euclidean 3-space. The conditions of coinciding Levi-Civita derivative, Fermi derivative and generalized Fermi derivative have been examined. Generalizations have been given for Euclidean 4-space and Euclidean n-space. In the last part of the fourth section, examples which are relevant to subject have been given. In the fifth section of the thesis, the considerations of the results have been given. Key Words: Fermi-Walker derivative, generalized Fermi-Walker derivative, generalized Fermi-Walker parallelism, generalized Fermi derivative, generalized non-rotating frame, Frenet frame, Darboux frame, Bishop frame, hypersurfaces.