The Surface Theory Of Hyperbolic Space

This paper mainly includes the curve and the surface theory in Euclidean space Rⁿ and hyperbolic space Hⁿ .We discuss how to study the surface'geometry by the method of moving frames and Lie group. There are three chapters in this paper:The first part is basic knowledge, it includes the local geometry theory of the curve and the surface and intrinsic geometry in 3-dim Euclidean space R³.In the second part, we generalize the curve and the surface'property to n-dim Euclidean space Rⁿ.Firstly, we give the Frenet formula of the curve in n-dim Euclidean space under general parameter and arc length parameter. Then we give two methods of constructing orthogonal frame by natural frame.Lastly, we derives the relation of Gauss curvature and mean curvature between the surface and its parallel surface with three methods ,which can be generalized to the calculation of variations and the curvature relation between the hypersurface and its parallel hypersurface in higher dimensional Euclidean space.In the third part, we combine the moving frames method with Lie group, define another metric following Euclidean space Rⁿ, and give hyperbolic space Hⁿ .On the basis of the study of Shi Shuguo in Shandong University, we discuss its surface and study the metric of this space. Then we identify the hyperbolic space Hⁿ with the upper half space of Euclidean space Rⁿ, and study them by the method of moving frames. We consider the group structure in hyperbolic space Hⁿ introduced by Shi Shuguo, and give a simple reasoning process of the surface's structural equation andmovement equation. Then we discuss two Gauss maps in hyperbolic space Hⁿ --normal Gauss map and hyperbolic Gauss map, and give a relation between them. Finally, we give another metric and obtain de Sitter space S₁ⁿ (1) .Then we can do a similar research to de Sitter space S₁ⁿ(1).