Constant Slope Surfaces In The Four-dimensional Euclidean Space

Since B.Riemann’s inaugural speech on "Basic assumptions about geometry" in 1854,Riemann geometry has become a very important basic theory in mathematics.The geometric properties of Riemann submanifolds are the focus of mathematicians.In this paper,we study a special submanifolds in four-dimensional Euclidean space--constant slope surfaces.A constant slope surface in Euclidean 4-space is a surface whose the position vector of a point of the surface makes constant angle with the normal plane at the surface in that point.In this paper,the parallel normal vector,the parallel mean curvature vector and the minimal constant slope surfaces are classified respectively.The thesis consists of five parts:The first chapter introduces the background and main research contents.Chapter 2 introduces the basic knowledge related to this paper.In chapter 3,the position vector of the constant slope surface at any point in the four-dimensional Euclidean space is first decomposed into the tangent part and the normal part.Then,the formulas of Gauss and Weingarten are used to obtain the basic quantities of the constant slope surface,including the Weingarten operator,the Levi-Civita connection and the normal connection.In chapter 4,the Schwarz equality and the basic equations of submanifolds are used to further consider the differential equation of an isometric immersion of a constant slope surface in Euclidean four-dimensional space,and simplify the equation by adding special conditions:normal vector fields are parallel,the mean curvature vector is zero,and the mean curvature vector is parallel.The last chapter summarizes the work of this paper.