Characterizations of darboux vector based slant helices

In this thesis focuses on the investigation of space-like curves defined in Minkowski four-dimensional space under the Darboux frame and on the characterization of their slant helix properties. The main objective of the study is to adapt the classical concept of slant helices, originally defined in Euclidean space, to Minkowski 4-space and to reveal the differential geometric properties of the resulting structures. Throughout this study, all curves under consideration are assumed to be unit-speed space-like curves parametrized by arc length. The thesis consists of four main chapters. In the first chapter, the motivation of the study and a review of the related literature are presented; the fundamental structure of Minkowski space, the classification of curves as space-like, time-like, and light-like, as well as the Frenet frame and curvature concepts are discussed in detail. In the second chapter, the Darboux frame for space-like curves in Minkowski 4-space is defined, the associated Darboux vector fields are derived, and their geometric interpretations are explained. Moreover, the mathematical foundations of k -type and (k,m) -type slant helices are established in this chapter. In the third chapter, which contains the main results of the study, the necessary and sufficient conditions for space-like curves to be k -type and (k,m) -type slant helices are derived. In addition, first-order nonlinear differential equations based on Darboux vectors are obtained and discussed together with theoretical findings, examples, and graphical illustrations. In the fourth and final chapter, the obtained results are evaluated in general terms, the contributions of the study to the existing literature are emphasized, and suggestions for future research are presented. Keywords: Slant helix, Darboux vector, space-like curve, Frenet curve, nonlinear equation