Parallel Weingarten surface in Euclidean space with density

This study consists of five sections. The first part contains literature review and fundamental facts. In the second section, the theory of curves in dimensional Euclidean space, the characteristic features of surfaces and the definitions and theorems about surfaces are reviewed. In the third section, the characteristic features of surfaces in density space were discussed while Gaussian curvature and mean curvature of the surface were calculated. Related theorems and results are expressed. The fourth chapter generates the original part of this study and it is divided by two subsections. In the first subsection of this chapter, the condition for a ruled surface to be a Weingarten surface in density with space has been studied, related theorems and results are shown. In the second subsection, the Gaussian and mean curvature of the parallel surface of a ruled surface in a density with space was calculated and the Weingarten condition of is examined, the related theorem, result and examples are given. In the final section, an extensive summary of the entire work was made and suggestions for future research on the condition of the parallel surface of a ruled surface to be Weingarten surface have been made.