Bertrand curves in spherical and hyperbolic space

This thesis consists of five chapters. In the first chapter, the literature survey about the concept of Bertrand curves in different ambient spaces is given. Then, we give needed fundamental definitions and theorems on curves theory in Euclidean, Minkowski, spherical and hyperbolic space, respectively. In the second section, the existing definitions and theorems about the concept of Bertrand curve in the spherical space. In the third section, we study the notion of (1,3)-Bertrand curve and its specifications in four-dimensional Euclidean space. Moreover, the theorems about relationships between these curves are given. In the fourth section, we review in detail the existing definitions and theorems about the concept of Bertrand curves in hyperbolic space. In the last part which is the original part of the thesis, we define (1,3)-Bertrand curve according to the causal character of (1,3)-normal plane of non-degenerate special Frenet curves in four-dimensional Minkowski space. Then, we give the needed condition that is the timelike (1,3)-Bertrand curve in four-dimensional Minkowski space for a non-planar Bertrand curve with non-constant curvature in hyperbolic space. However, we give methods of obtaining Bertrand curve in hyperbolic space by the spacelike or timelike (1,3)-Bertrand curve in four-dimensional Minkowski space and obtain relations between curvatures of these curves for the first time in this thesis. Finally, we show that a helix is also a Bertrand curve in hyperbolic space and draw images of the curve and its Bertrand mate in Poincare ball model of hyperbolic space as an example.