The Generalized Ruled Surfaces In 3-Euclidean Space

Differential geometry is a subject which has a long history. The moment of the birth of calculous is also that of the birth of differential geometry. 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. At the same time, this subject is always being update in the contents and the methods.The definition of the ruled surfaces is promoted, and the definition of the generalized ruled surfaces is given in this paper. Correspondingly the definitions of three kinds of special generalized ruled surfaces (parabolic surface,elliptic surface,hyperbolic surface) are given too.In this paper, parabolic surface,elliptic surface,hyperbolic surface with special conditions (the tangent directions which are vertical to the tangent directions of v-curves of these surfaces are fixed) are discussed. The parabolic surface with constant mean curvature,constant Gauss curvature and satisfying certain condition between mean curvature and Gauss curvature is studied too.