Ruled And Translation Surfaces In Three-Dimensional Space

In1827, Gauss published a book "General investigation of curved surfa-ces". In this book, he set up the internal geometry of surfaces. The main id-ea was to emphasize some properties of the surfaces, which depend only on the first fundamental form. His examination as the following:the length of curves on the surfaces、the angle between two curves、the surface area of a region、geodesic、geodesic curvature and the total curvature and so on. Thus, the study of surfaces became a hot issue of differential geometry.Considering the curves on the ruled surfaces, Shyuichi Izumiya discuss-ed cylindrical helices and Bertrand curves in3-dimensional Euclidean spac-e. The principal normal surface has been studied in this article,(a) Let the base curves be Mannheim curves, we can get the following two conclusions:(1)the geodesic line and the striction line are the Mannheim companion line-s;(2)the minimal locus is Mannheim curve itself or r(s)=r(s)+(2λ-1)β.(b) When the base curves are Bertrand curves, we have:(1)the minimal locus is an Bertrand curve or its companion line;(2)the non-linear asymptotic curve family is|τ|=C(1-λ)2.(c) If the base curves are general helix, the conclusion is:the ratio of the two principal curvature functions is a constant, when the torsion is τ-1=Cls+C2. The principal curvature is the curvature along the center of its circle of curvature,(d) Let the base curve be one of asymptotic curves of the surface, the ratio of the two principal curvature functions is a constant. The principal curvature is the curvature along the asymptotic curves.In this paper, we also study some surfaces in the Minkowski space. A surface is called a translation surface, if the surface can be written as z=f(x)+g(y).y=f(x)+g(z) or x=f(y)+g(z) According to the direction in Minkowski space, the translation surfaces can be considered as the following six types:(1) along spacelike direction and spacelike direction;(2) along spacelike direction and timelike direction;(3) along lightlike direction and lightlike direction;(4) along lightlike direction and spacelike direction;(5) along timelike direction and lightlike direction;(6) along timelike direction and timelike direction. Then the translation surfaces are called type1～type6surfaces, respectively. H.L.Liu had classified the type1and type2translation surfaces, which had constant mean curvature and zero Gaussian curvature.The fourth parts of this paper is the classifications of translation surfaces. For the type3, we choose the pseudo-orthogonal frame. We use the orthogonal frame to study type5or type6. Let E13be the Minkowski3-space with the inner product {x,y)=x1y1+x2y2-x3y3. Translation surface Sa of type5and type6can be written as x(u,v)={f(u+av)+g(v),u,v}, and(i)when|a|=1,Sa is type5translation surface;(ii)when|a}>1, Sa is type6translation surface. To simplify the solving process and avoid the partial differential equations, we select the non-degenerate transformation At last, we get the Weingarten-type classification theorems.L.K.Graves gave the definition of B-scroll in1979. But there are few conclusions about properties of B-scroll surfaces. So it is meaningful to study the properties of B-scrolls. By using the cone curve theory, H.L.Liu gave some characterizations and examples of B-scroll surfaces. In our paper, using Pitch function of ruled surfaces, we give a new characterization of B-scroll. The characterization of the non-developable ruled surface can be gotten by Pitch function and some special frames in3-dimensional space. In the Euclidean space, we select spheral Frenet frame. In the Minkowski space, we have de Sitter Frenet frame with spacelike ruling, hyperbolic Frenet frame with timelike ruling and cone Frenet frame with lightlike ruling. Finally, we obtain that a surface is a B-scroll if and only if its Pitch function vanishes identity.