Helicoidal Surfaces With Pointwise Gauss Map In Minkowski 3-space

As an important geometrical space, Minkowski space is a new field relating to Euclidean Space which is familiar to us. Therefore, it is meaningful for us to study the corresponding curves and surfaces in this Space. However, due to the instability of metric, some problems which seem easy and reasonable become complex in Euclidean Space.In local theory of the surfaces in differential geometry, there is a kind of problem called corresponding relationship between the surfaces, which is generally considered as Gauss map on the surfaces meeting certain correspondence. Some surfaces with finite type Gauss map and one by one Gauss map have been discussed in 3-dimensional Euclidean and Minkowski Space. This paper is carried out under the idea and discussed 5 kinds of helicoidal surfaces. In the preliminary, we introduce the creation, development, studying contents and methods of differential geometry. In Chapter One, we introduce some basic knowledge, such as Minkowski Space,5 cases helicoidal surface, Gauss map on surface, the definition of Laplacian and so on. In Chapter Two, we introduce several kinds of special helicoidal surfaces. In Chapter Three and Four, we analyze normal polynomial and rational helicoidal surfaces respectively. The last chapter is the summary and expect of the whole paper.