Centroaffine Surfaces And Centroaffine Immersions Of Codimention Two

Felix Klein’s1872Erlangen program puts geometry down to the geometric invariant theory of the transitive transformation group, and then categorizes them. Thus, there is a transitive transformation group, there is a geometry attached to this transitive transformation group. In affine space An, the vector v can be transformed to the vector v. If there exists a relation between v and v satisfying that v=Av, where A is an×n non-degenerate matrix, all transformations compose a group and this group is called as a centroaffine transformation group. Centroaffine geometry of affine space An is a branch of discussing invariance of graphics under centroaffine transformation, and it belongs to centroaffine transformation group of An. This paper considers the geometry attached to centroaffine transformation group, that is, geometric invariance of submanifolds under centroaffine transformation group G in Rn+1(or Rn+2).If the immersions x:Mâ†'Rn+1(or x:Mâ†'Rn+2) always keeps position vector fields x transversal to tangent plane x*(TM), there exists a symmetric2-forms g with regard to centroaffine transformation group G, and g is called as the centroaffine metric. For centroaffine immersions x:Mâ†'Rn+1, Professor Wang Changping calculated the first and the second Variational formula of centroaffine hypersurfaces with respect to g, and obtained a new centroaffine invariant---Tchebychev operator. Professor Liu Huili and Professor Wang Changping further study of this Tchebychev operator, and got the classification of Tchebychev sufaces (Tchebychev hypersurfaces). From then on the classification of hypersurfacs became one of the important works of studing centroaffine immersions x:Mâ†'Rn+1, and the methods of the classification of hypersurfacs are mainly based on the centroaffine invariant and the relationship between them. Since there is not metric in ambient space Rn+1of centroaffine immersions, the calculation in ambient space Rn+1is very inconvenient. On the other hand, the complex relationship between centroaffine variants and wide range of centroaffine hypersurfaces make the work of the classification of centroaffine hypersurfacs difficult to carry on. In response to these situations, this paper poses a solution:firstly the all centroaffine hypersurfacs are divided into several broad categories, and then we get detail classifacation from the larger category. This paper considers two major categories of centroaffine hypersurfacs-centroaffine translaton surfaces and centroaffine ruled surfaces. In the third part of this paper, we get basic structure equations of centroaffine translation hypersurfaces and some centroaffine invariants. According to these equations and relational partial differential equation, we obtain centroaffine translation surfaces with constant Gauss curvature, centroaffine translation surfaces with constant Pick invariant, and centroaffine translation surfaces with||T||2=constant in R3. On the other hand, by solving certern partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-affine space R3and prove that a nondegenerate centroaffine ruled surface is centroaffine minimal if and only if its Gauss curvature is constant.For centroaffine immersions of codimension two x:Mnâ†'Rn+2, there always are two different research methods. Nomize and Sasaki defined a pre-normalized Blaschke vector field ξ using the conditiontrh{T(X,Y)+h(SX,Y)}=0, where h is the centroaffine basic form, S is the Weingarten operator, T is centroaffine2-form. On the other hand, Professor Liu has studied centroaffine immersions of codimension two using a new method to define the metric g, and chose Δgx as the second transversal vector fields, where Δg is the Laplacian of metric g. When we study centroaffine immersions of codimension two, ξ and Δgx all can be think as the second transversal vector field, and the relations and differences between them are the questions we must consider. In the fourth part of this paper, we calculate the basic equations and invariants centroaffine immersions of codimension two using moving frames. Depending on these equations and invariants we get the relation between these two second transversal vector fields, that is,ξ=1/nΔgx-H/2x, where H is centroaffine mean curvature. Then we compute the first and also second variational formulas of centroaffine volume integral using Δgx as the second transversal vector field and then define the minimal centroaffine immersions of codimension two, which is consistent to the minimal centroaffine immersions of codimension two using ξ as the second transversal vector field. As some examples, we proof that homogeneous surfaces with vanishing Pick invariant in R4are centroaffine minimal.In this paper, we introduce the history of Differetial Geometry, especially about Affine Differetial Geometry (ADG) and Centroaffine Differetial Geometry in the first part. In the second part we describe some knowledge in Differetial Geometry, which is the basic of Centroaffine Differetial Geometry. Then in the third part we talk about centroaffine hypersurfaces. Mainly, we get some results about centroaffine translation surfaces and centroaffine ruled surfaces in R3. In the last part we introduce the centroaffine subminafolds of codimention two. Here we compute the first and the second formulas of area variantion and get the relation between different centroaffine normalizations.