中心仿射微分几何若干问题研究

项目名称： 中心仿射微分几何若干问题研究

项目编号： No.11201056

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 杨云

作者单位： 东北大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 仿射微分几何；中心仿射变换群；幺模仿射变换群；曲率；子流形

英文摘要： 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 pro

英文关键词： affine differential geometry；centroaffine differential geometry；equiaffine transformation group；curvature；submanifolds