Curvature and Symmetry (B01)

流形的几何性质在多大程度上决定了它的全局拓扑，这是整体微分几何中的经典问题。在第一个子项目中，我们研究具有环面对称性的正曲线流形的拓扑。我们认为，这个子项目中使用的方法也可以用来攻击正四元数Kähler流形的Salamon猜想。在第三个子项目中，我们研究非负曲线流形的基本群。另外两个子项目涉及流形的分类，所有的测地线都是闭的，以及黎曼测地线上闭测地线的存在。

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.