The Ruled Surfaces In Pseudo-Euclidean Space And Euclidean Space

In this thesis the ruled surfaces in pseudo-Euclidean space and Euclidean space are studied. Firstly, by means of moving frame, some properties of the ruled surfaces in pseudo-Euclidean space are obtained, including the ruled surfaces being minimal, totally developable, totally geodesic and totally umbilical. Some equivalent conditions in which the ruled surfaces are totally developable are obtained. It is obtained that k +1-dimensional ruled surfaces in pseudo-Euclidean space are totally geodesic if and only if they are minimal and totally developable. Specially, when k ≥ 2 and the generating spaces are space-like or time-like, they are totally geodesic if and only if they are totally umbilical. Gauss - Kronecker curvature of n-dimensional ruled hypersurfaces in pseudo-Euclidean space vanishs providing n≥3. But the ruled surfaces in E³ or E₁³ have vanishing Gauss curvature if and only if they are developable. Finally, the 2-dimensional ruled surfaces in 4-dimensional Euclidean space are studied. By means of moving frame, it is obtained that the helicoids and planes are only the minimal 2-dimensional ruled surfaces in 4-dimensional Euclidean space.