一类闭半黎曼流形的几何与拓扑性质

Projective equivalent metric is the main content in the study of differential geometry, and it has important potential applications in general relativity, mechanical system and so on. Up to now, projective equivalent metric has been studied extensively. The traditional classification method of projectively equivalent metric is tensor analysis, this method is only applicable to the local neighborhood of points in sem-Riemann manifold, it fails to derive the global projective equivalent metrics; and some improved methods of tensor analysis may get certain global projectively equivalent metrics but under the condition of imposing some additional curvature conditions, thus do not apply for the general case..This topic proposed on the basis of the theory of integrable geodesic flow, combined with tensor analysis and nonlinear analysis, aims to study the global geometrical and topological properties of the closed semi-Riemannian manifold admitting projective equivalent metric. Firstly, we study 2-dimensional closed semi-Riemannian manifold and give the global classification of nontrivial projective equivalent metric; then we research on projective Lichnerowicz-Obata conjecture in semi-Riemannian case; Finally, we study the obstruction for a semi-Riemannian manifold admitting global projective equivalent metric. These studies are expected to further clarify the properties of projectively equivalent metrics on semi Riemann manifold and develop the applications of projectively equivalent metrics in the theory of relativity. Therefore, it is necessary to carry out the research of projective equivalent metrics.

射影等价度量的分类问题属于微分几何的经典问题，且在广义相对论、力学系统有重要的应用，目前国内外几何学家、物理学家对此展开了较为充分的研究。现阶段射影等价半黎曼度量分类的方法仅限于张量分析，此法只适用于半黎曼流形上点的局部邻域内，不能得到整体分类；而一些改进的张量分析法也是在强加了曲率条件后才得到射影等价度量的大范围分类，难于运用到一般的半黎曼流形。. 本课题拟运用可积测地流的理论，在传统的张量分析研究的基础上，结合非线性分析旨在研究拥有射影等价度量的闭半黎曼流形几何与拓扑等大范围的性质。首先给出闭半黎曼流形上不平凡的射影等价度量的整体分类；其次研究半黎曼情形下的射影Lichnerowicz-Obata猜想；最后深入分析一般的半黎曼流形拥有整体射影等价度量的条件及个数问题。这些研究可望进一步弄清半黎曼流形上射影等价度量的性质以及发展射影等价度量在相对论中的应用。因此，开展这一方面的研究

射影等价度量的分类问题属于微分几何的经典问题，且在广义相对论、力学系统有重要的应用。本项目通过可积测地流中大范围积分的构造，给出闭半黎曼流形上整体的射影等价度量的构造，从而实现闭半黎曼流形上射影等价度量的完全分类，解决了Beltrami问题；利用闭曲面上二次可积测地流存在性的拓扑障碍研究并给出2维半黎曼情形下的射影Lichnerowicz- Obata猜想：(M, g)是一个闭的连通的 2维闭半黎曼流形，假设一个连通李群G通过射影变换作用于M，则该李群G是等距作用的或者g是常曲率的；最后给出闭的半黎曼流形拥有射影等价度量的障碍分析。