Curvature and Symmetry (B01)

流形的几何性质在多大程度上决定其全局拓扑是全局微分几何中的一个经典问题。在第一个子课题中，我们研究了环面对称的正弯曲流形的拓扑结构。我们认为在这个子项目中使用的方法也可以用来攻击Salamon猜想的正四元数Kähler流形。在第三个子课题中，我们研究了非负弯曲流形的基本群。另外两个子课题是关于所有测地线都是闭合的流形的分类和黎曼轨道上闭合测地线的存在性。

The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. In a first subproject we study the topology of positively curved manifolds with torus symmetry. We think that the methods used in this subproject can also be used to attack the Salamon conjecture for positive quaternionic Kähler manifolds. In a third subproject we study fundamental groups of non-negatively curved manifolds. Two other subprojects are concerned with the classification of manifolds all of whose geodesics are closed and the existence of closed geodesics on Riemannian orbifolds.