Geometry and Topology of Riemannian Manifolds

DMS-0204671.首席研究员计划继续他在微分几何和相关领域中的全球问题的工作。将特别强调通过额外的结构追求全球黎曼几何。这包括但不限于由于对称性的存在而产生的结构，以及由于采用外部或内部限制而产生的结构。我们关于曲率、对称性和拓扑之间关系的研究工作是由目标提出的程序指导的：`分类或描述具有正曲率或非负曲率和大等距群的流形的结构。这一领域目前正在各个方向取得重大进展。该项目还将包括对流形在下曲率下界下崩溃所产生的结构的调查，以及对有限度量空间中各种空间的结构的调查所产生的度量不变量的调查。一方面，在许多几何情况下，从一开始就不存在对称性，但从几何学中很容易出现对称性。这种情况包括刚性问题和有界曲率下的坍塌问题。然后，通过所提议的工作所获得的一般结果可以被引用并帮助解决不存在对称性的原始问题。这种方法已经成功地使用了。另一方面，提出的工作提供了一种系统的方法来寻找具有正曲率或非负曲率的空间的新例子，这可以说是当今全球黎曼几何面临的最核心和最困难的问题之一。

ABSTRACT DMS - 0204671.The principal investigator plans to continue his work on global problems in differential geometry and related areas. Special emphasis will be devoted to the pursuit of global Riemannian geometry via additional structures. This includes but is not limited to structures arising from the presence of symmetries, and to structures arising from taking external or internal limits. Our efforts concerning investigations of relations between curvature, symmetry and topology is guided by the program set forth by the aim: ``Classify or describe the structure of manifolds with positive or nonnegative curvature and large isometry groups". This area is currently experiencing significant advances in various directions. The project will also include investigations of structures arising from the collapse of manifolds under a lower curvature bound, and of metric invariants coming from investigations of the structure of various spaces of finite metric spaces.The significance of the main part of the proposal is twofold. On the one hand there are many geometric situations where no symmetries are present from the outset, but where symmetries emerge non-trivially from the geometry. Among such situations are rigidity problems and collapsing problems under bounded curvature. General results achieved through the work proposed can then be invoked and help solve the original problem, where no symmetries were present. This method has already been used successfully. On the other hand, the proposed work provides a systematic approach for finding new examples of spaces with positive or nonnegative curvature, arguably one the most central and difficult issues facing global Riemannian geometry today.