Bour theorem and GAUSS map in Minkowski space-time

In this study; Bour's theorem for helicoidal surfaces in Minkowski space-time is studied and Gauss maps of isometric surface pairs are examined. This thesis consists of four main parts. In the first chapter, the importance of rotational surfaces and the importance of rotational surfaces and helicoidal surfaces which are the generalization of them are explained. In addition, the results in the literature related to Bour's theorem are mentioned. In the second chapter, some basic concepts about curves and surfaces in Minkowski space-time used in this thesis are given. In the third chapter, isometric rotational surfaces according to Bour's theorem are obtained for three types of spacelike helicoidal surfaces in Minkowski space-time. Under the condition that the isometric surfaces have same Gauss map, the explicit parametrizations are obtained. In addition, the reasons for the different Gauss maps of some isometric surfaces are given. Specifically, it is studied on right helicodal surfaces. In addition, it is shown that the Gauss curvature of the isometric surfaces according to Bour's theorem is the same. Finally, concrete examples of isometric surfaces with same Gauss map are given and the graphs of the surfaces were drawn with the Wolfram Mathematica 10.4 program. In the fourth chapter, similar to the third chapter, four types of timelike helicoidal surfaces in Minkowski space-time are studied. However, in Minkowski space-time, because timelike helicoidal surfaces can have a spacelike or timelike profile curve, different results are obtained in this section than in the third section. In the fourth chapter, similar to the third chapter, four types of timelike helicoidal surfaces in Minkowski space-time are examined. In Minkowski space-time, however, different results are obtained in this chapter than in chapter three, as timelike helicoidal surfaces can have a spacelike or timelike profile curve.