Geometry and Topology in the Presence of Lower Curvature Bounds

作为经典欧几里得几何和球面几何的自然扩展，非负曲率或正曲率流形几何自整体黎曼几何开始以来一直发挥着核心作用。这个角色在过去的几十年里被放大了，因为非负或正曲率的空间在相当普遍的情况下自然出现，包括极限过程。在这种一般性中，正弯曲空间（直到缩放）与单位球对光滑黎曼流形的作用完全相同。我们对低维非负弯曲空间的理解也在最近著名的庞加莱猜想和几何化猜想的解决中发挥了关键作用。在高维中，对于流形或非负曲率或正曲率的空间，一般知之甚少。也只有少数结构和少量的例子是已知的。由于所有已知的例子都来自群构造，并且具有相当大的对称群，因此本提案的主要目的之一是通过描述或甚至分类具有大对称群的流形来扩展我们对具有正曲率或非负曲率的流形的理解。这个结合几何、拓扑和表示理论的程序已经获得了相当大的动力，并产生了几个分类结果，以及许多新的非负曲率流形的构造，以及新的有希望的正曲率候选者。球面、欧几里得空间和双曲空间正是（单连通）空间，其特征是具有恒定曲率和极大对称群。比这些空间更弯曲的空间在几何上的特征是测地线三角形（边长最短的三角形）比常曲率空间“更胖”。例如，如果空间中的测地线三角形比欧几里得平面（其角度之和为180度）“更胖”，则空间具有非负曲率。这种空间在几何中起着基本的作用，是经典黎曼几何的延伸，它处理这种类型的光滑和规则空间。正曲率、非负曲率，甚至“几乎非负”曲率都起着特殊的作用，对它们的研究对它们都是至关重要的。在物理学的许多方面，我们的目的是分析并最终描述存在大量对称性的正弯曲空间和非负弯曲空间（如上面的经典常曲率模型空间）。这些研究还将为分析“几乎非正弯曲空间”提供“模型”，从而为具有低曲率界的所有空间的结构提供新的见解，并可能通过极限过程产生对非负曲率流形的一般长期追求的限制。

As natural vast extensions of the classical Euclidean and spherical geometries, geometry of manifolds with non-negative or positive curvature has played a central role since the beginning of global Riemannian geometry. This role has only been amplified in the last few decades since spaces with non-negative or positive curvature arise naturally in quite general contexts, including limit processes. In this generality, positively curved spaces (up to scaling) play exactly the same role as unit spheres do to smooth Riemannian manifolds. Our understanding of low dimensional non-negatively curved spaces also played a pivotal role in the recent solution of the famous Poincare and geometrization conjectures. In higher dimensions relatively little is known in general about manifolds or spaces with non-negative or positive curvature. Also only a few constructions and a modest number of examples are known. Motivated by the fact that all known examples come from group constructions and have fairly large groups of symmetries, one of the primary aims of this proposal is to expand our understanding of manifolds with positive or non-negative curvature by describing or possibly even classifying those with large symmetry groups. This program which combines geometry, topology and representation theory has already gained considerable momentum, and has resulted in several classification results as well as in the construction of many new manifolds with non-negative curvature, and new promising candidates for positive curvature.The sphere, the Euclidean space, and the hyperbolic space are exactly the (simply connected) spaces characterized by having constant curvature and also by having maximal symmetry group. Spaces being more curved than these spaces are characterized geometrically by the property that geodesic triangles (triangles with shortest side lengths) are "fatter" than in the constant curvature space. For example a space has non-negative curvature if geodesic triangles in the space are "fatter" than in the Euclidean plane (where the sum of angles is 180 degrees). Such spaces play a fundamental role in geometry and form an extension of classical Riemannian geometry, which deals with smooth and regular spaces of this type. The ones of positive, non-negative curvature, or even "almost non- negative" curvature play a particular role and their investigations are essential to all of them. As in many part of physics our purpose in this proposal is to analyze and ultimately describe positively curved spaces and non-negatively curved spaces where large groups of symmetries are present (as is the case for the classical constant curvature model spaces above). These investigations will also provide "models" for analyzing "almost non-positively curved spaces" and thereby give new insights to the structure of all spaces with a lower curvature bound and possibly yield general long sought after restrictions on manifolds with non-negative curvature via limit processes.