Curves And Surfaces In Affine Space

Affine geometry of affine space A" is a banch of talking affine invariant and affine invariance. Let A3be3-dimensional space with the semi Euclidean inner product <x,y>λ=λ1x1y1+λ2x2y2+λ3x3y3, where λ1=±1, λ2=±1, λ3=±1and x=(x1,x2,x2), y=(y1,y2,y3)∈A3, this space is called semi Euclidean space. When λ1=λ2=λ3=1, this semi Euclidean space A3is the3-dimensional Euclidean space E3. When λ1=λ2=1and λ3=-1, this semi Euclidean space A3is the3-dimensional Minkowski space E3(or Lorentzian space L3). It is the Lorentzian space that provides with the effective mathematical method to Einstein who founded and extended the theory of relativity. Also, the theory of relativity accelerates the study of Lorentzian space. So it is useful to study curves and surfaces in the semi Euclidean space.In this paper, we will study curves and surfaces in3-dimensional affine space. A characterization of the ellipse and hyperbola had been studied by Dong-Soo Kim and Young-Ho Kim. Here in the section of curves, firstly, we will study a characterization of parabola in2-demensional Euclidean space, using the support function h and the curvature k of the parabola. We can get that if the support function and the curvature of a curve satisfy the equation κ=-p2/8h3, the curve is a parabola whose focus is at the origin. Next we get that the relationship between the affine curvature κ1and the Euclidean curvature k for a curve in2-dimensional space isAs we known, a curve is a plane quadratic curve if and only if the affine curvature is a constant. Using this conclution we will characterize plane curves in the Euclidean space. Thirdly, considering a curve as the curve in3-dimensional Euclidean space and also in3-dimensional Minkowski space, we will get some invariants of the curve, such that κ2τ(ds)6and so on, which are independent of the choice of inner products. In the section of surfaces, there have three parts. The first part will study factorable minmal surfaces in3-dimensional Euclidean space and3-dimensional Minkowski space. As a surface can be regarded as a graph locally, tbe factorable surface can be written as z=f(x)g(y), y=f(x)g(z) or x=f(y)g(z),By solving equation, we class the factorable minmal surfaces. In the second part, we will study surfaces of revolution with prescribed mean curvature in3-dimensional Minkowski space. Since there are three kinds of rotations in3-dimensional Minkowski space, we have three kinds of rotation matrices respect to spacelike, timelike and lightlike rotation axes. In this paper, we will class these rotation surfaces with the prescribed mean curvature by solving some differential equation. In the last part, by solving certain partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-dimensional affine space and prove that a non-degenerate centroaffine ruled surface is centroaffine minimal if and only if its Guass curvature is constant.