Tube surfaces with q-frame in 4-dimensional space

This thesis aims to investigate a special type of surface in 4-dimensional Euclidean space known as the tube surface with an alternative structure called q-frame. The study is divided into five chapters. The first chapter provides an overview of the thesis content and highlights the significance of the study. The second chapter reviews prior research on canal surfaces, with particular attention to a specific type of canal surface known as tube surface. The second chapter also includes a presentation of research findings of tube surfaces and canal surfaces relevant to this thesis. The third chapter introduces fundamental definitions, theorems and derivative equations related to Frenet and q-frames for a space curve in 3-dimensional and 4-dimensional Euclidean spaces to enhance the comprehension of the thesis topic. Subsequently, information about surfaces in 4-dimensional and n-dimensional Euclidean spaces is given. The general parametrization of the tube surface in this space is expressed according to the Frenet frame. In the fourth and fifth chapters, the original calculations are carried out for tube surfaces with Frenet and q-frame, respectively. Using the general parametrization of the 2-dimensional tube surface with Frenet and q-frame vectors, unit normal vector fields U1 and U2 are obtained in 4-dimensional space. Additionally, shape operator matrices with unit normal vector fields are computed through the fundamental forms of the tube surface. The algebraic invariants of these shape operators, namely principal curvatures, Gauss curvatures, and mean curvatures, are derived. On the other hand, the characteristic properties of the surface such as the Christoffel symbols, Gaussian curvatures and mean curvatures of the 2-dimensional tube surface are also calculated using the coefficients of the fundamental form. Finally, examples of 2-dimensional tube surfaces according to Frenet-frame and q-frame in 4-dimensional Euclidean space are studied, and the curvatures of the surfaces are examined. The shapes of these tube surfaces, along with their central curves in the projection spaces, are illustrated. Key Words: Frenet frame, q-frame, tube surface, curvatures.