Surfaces Of Revolution With Finite Type Gauss Map

With the development of mathematics, the basic theory of mathematics has become more and more deep and perfect. At the same time, it has also impelled the huge transform method in mathematical research. As the base of all science, mathematics is breaking the traditional aspects and is seeking all areas in human knowledge. As a science of describing physical astrospace, geometry also better reflects different aspects and areas. Now, we not only study mathematics in flat space, but also develop it in curved space. At present, the most suitable curved space is manifold. One kind of special manifold is surface. The notion of finite type submanifolds in Euclidean or pseudo-Euclidean space has become a useful tool for investigating and characterizing many important submanifolds. The notion of finite type was extended to differential maps, in particular, to Gauss map of submanifolds. Then we have the conception of pointwise finite type Gauss map.In this thesis we mainly discuss the problem of surfaces of revolution with finite type Gauss map. Especially, we study the rational surface of revolution with finite type Gauss map systematically and completely. Furthermore, we obtain the necessary and sufficient conditions for the rational surface of revolution with finite type Gauss map. These are the development of the conclusions of the surface of revolution with pointwise 1-type Gauss map. In Chapter Three, we obtain the formulas of pointwise n- type Gauss map for plane, circular cylinder and right cone respectively.