Geometry and Topology of Riemannian Manifolds

摘要DMS -0204671.首席研究员计划继续他在微分几何和相关领域的全球问题方面的工作。特别强调将致力于追求全球黎曼几何通过额外的结构。这包括但不限于由对称性的存在而产生的结构，以及由外部或内部限制而产生的结构。我们的努力有关调查之间的关系曲率，对称性和拓扑结构的指导下提出的计划的目的："分类或描述结构的流形与积极或非负曲率和大型等距群”。这一领域目前正在各个方面取得重大进展。该项目还将包括调查的结构所产生的崩溃的流形下的一个较低的曲率界，和度量不变量来自调查的结构的各种空间的有限度量spaces.The意义的主要部分的建议是双重的。一方面，在许多几何情形中，从一开始就不存在对称性，但对称性却从几何中非平凡地出现。在这样的情况下是刚性问题和有界曲率下的崩溃问题。 通过所提出的工作取得的一般结果可以被调用，并帮助解决原来的问题，其中没有对称性。该方法已成功使用。另一方面，拟议的工作提供了一个系统的方法来寻找新的例子，空间的积极或非负曲率，可以说是一个最核心的和困难的问题，今天面临的全球黎曼几何。

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.