Some Un-developable Ruled Surfaces In Euclidean 3-space

Differential geometry is a subject which has a long history. However, the life force of this subject is still blooming until now. It has a deeper influence on other branches of mathematics in recent years, and expends its range of influence to other subjects more and more. Curve theory and surface theory are two important elements in differential geometry. Ruled surface which has good characters occupies very important position in surface theory. In 3-dimensional Euclidean space ruled surface can be developable surface, or un-developable surface by its Gaussian curvature.The characters of the developable surface are clear until now, but the study of the un-developable surface is rare.We denote the Euclidean 3-space by E3 and a regular parameter surface with the parameters u and v in E3 by r(u,v)= a(u)+vb(u). At first, let r(u,v)= a(u)+vb(u) be a un-developable ruled surface in E3 and b2(u)= 1, the parameter u is the arc length parameter of b(u) as a unit spherical curve in E3. Furthermore, we assume that the base curve a(u) of the ruled surface r(u,v) is the striction line of the surface that means 〈a’(u),b’(u)〉= 0, here we use a’=da/du■. Then we calculate the Frenet formula and express a’(u) the by it. In this paper, we let kg= c, we can get the function as a’(u)= (λ+Cμ)x+c1μ, we get some types of un-developable surface by discussing the parameters, when they are (λ+cμ)= 0, (μ+cμ)= por(λ+cμ)= p(u). Finally we can study the characters of these un-developable surfaces, such as the Gauss curvature, mean curvature, the first and second fundamental forms.