Directional ruled hypersurfaces in 4-dimensional space

This thesis aims to investigate ruled surfaces generated by quasi-vectors in 4-dimensional Euclidean space using a quasi-frame. Our study consists of 5 chapters. In the first two chapters, Introduction and Literature Survey, historical works on curves, surfaces, ruled surfaces, Frenet, and q-frames are mentioned. In the third chapter, in differential geometry, curves are the keystone and surfaces are the other keystone. One of the structures where the concepts of curve and surface are used together is ruled surfaces. 2-dimensional ruled surfaces are surfaces formed by moving a line along a curve. In particular, in 4-dimensional space, given a directrix space that is a 2-dimensional subspace spanned by an orthonormal basis, a 3-dimensional surface along the curve formed by a unit speed curve and the 2-dimensional directrix space, is called a ruled hypersurface. In this study, Frenet vectors, Frenet differential equations, and the curvatures κ, τ and η of the curve with Frenet frame in 4-dimensional Euclidean space are first introduced. Then necessary concepts to study the differential geometric properties of hypersurfaces are given. In the fourth and fifth chapter, firstly, the parametric expression of the special ruled hypersurfaces formed by the ruled hypersurface stretched by the tangent vector T and normal vector N, which are units of Frenet vectors, and the planes stretched by binormals B1 and B2 vectors are given. Differential geometric calculations of this ruled hypersurface are obtained. In addition, parametric expressions and geometric properties of 2-dimensional directional ruled hypersurfaces generated by unit tangent and unit quasi-normal vectors denoted as T and Nq, respectively, instead of the rectifier vector in 4-dimensional Euclidean space are given. Moreover, for a suitable space curve, a parametrization of the Frenet frame and quasi-frame ruled hypersurface and examples of the calculation of its geometric properties are presented. Furthermore, to facilitate comprehension of the example in 4-dimensional space, the shapes of this ruled hypersurface have been graphed in 3-dimensional projection spaces. Key Words: Quasi-frame, ruled hypersurface, shape operator, curvatures