中心仿射微分几何若干问题研究

Transformation is a basic and important study objective in differential geometry, such as orthogonal transformation in Euclidean geometry, projective transformation in projective geometry and equiaffine transformation in affine geometry. So this program focuses on centroaffine transformation in affine space, which considers differential invariants and integral invariants under centroaffine transformation...The basic difference between Euclidean space and affine space is that there is no definition of metric in affine space, so the derivation of the induced metric is the key problem. In early stage, we introduced a centroaffine metric on submanifolds of centroaffine immersions by using some theories of differential geometry and differential equation, and got some original results about centroaffine submanifolds by initial studying centroaffine invariants and applying them in affine 3-space and affine 4-space...In this program, the task that needs to be finished first is to solve some ordinary and partial differential equations appearing in previous work and to find their geometric meaning. ..Then we could calculate more and deeper relations among those centroaffine invariants obtained from the original results. At the same time, the necessary work is to obtain some new centroaffine invariants and explore what properties these centroaffine invariants can determine in different centroaffine immersions. ..Furthermore, expanding those theorems in affine 3-space and affine 4-space to higher dimensional affine space can be done, which would generate more general theories...Last but not least, depending on the geometric meaning of those centroaffine invariants both in affine space and in Euclidean space, some classifications for centroaffine submanifolds would be significant and meaningful...In brief, the target of this program is to expand some results of differential geometry in Euclidean space to affine space.

变换是在几何研究中涉及到的重要而又基本的对象之一,如欧氏几何的正交变换，射影几何的射影变换，仿射几何的么模仿射变换等等。本项目研究涉及中心仿射变换，即研究仿射空间中子流形在中心仿射变换下的微分不变量和积分不变量。在仿射空间中研究子流形首先要解决诱导度量问题。在前期工作中，我们运用微分几何和微分方程的一些理论定义了子流形上的中心仿射度量，得到了一些基本结果。本项目首先在前期工作的基础上，解决已经出现的微分方程和与它们相关联的几何意义。其次将挖掘在前期研究工作中出现的这些中心仿射不变量之间的深入联系，同时发现一些新的不变量，探讨它们对不同类型中心仿射浸入的影响。再次需要将3维和4维仿射空间中的研究结果推广到高维仿射空间中，得到更一般的理论。最后将基于中心仿射不变量在仿射空间中和欧氏空间中的几何意义，对中心仿射子流形进行一些有意义的分类。本项目预期将欧氏空间微分几何的部分结果推广到仿射空间。

1872年德国数学家克莱因在埃尔朗根大学的教授就职演讲中，作了题为《关于近代几何研究的比较考察》的论文演讲，论述了变换群在几何中的主导作用，把到当时为止已发现的所有几何统一在变换群论观点之下，明确地给出了几何的一种新定义，把几何定义为一个变换群之下的不变性质。这种观点突出了变换群在研讨几何中的地位，后来简称为《埃尔朗根纲领》。因此，仿射微分几何的根本目的是研究子流形在仿射变换群下的不变性质。本项目研究了中心仿射变换下，仿射空间中的超曲面和余二维子流形的微分不变量和积分不变量。首先，本项目研究了3 维仿射空间中的仿射旋转曲面在中心仿射变换下的不变性质。研究Pick 不变量为零的中心仿射旋转曲面，得到3 维仿射空间中旋转曲面的分类。求解相关的常微分方程和偏微分方程，对具有关于中心仿射度量的切比雪夫常法化向量场的旋转曲面进行分类。其次，本项目研究了4 维仿射空间中的中心仿射曲面。得到了余二维中心仿射浸入的完美定理。在退化的第二基本形式和Pick 不变量为零的条件下，得到了平坦的中心仿射曲面的分类。在平行的第二基本形式和Pick 不变量为零的条件下，得到了平坦的中心仿射曲面的分类。