Geometry and Topology in the Presence of Lower Curvature Bounds

具有非负或正曲率的流形几何作为经典欧几里得几何和球面几何的自然推广，自全球黎曼几何诞生以来一直扮演着中心角色。这种作用只是在最近几十年才被放大，因为具有非负曲率或正曲率的空间在相当一般的背景下自然产生，包括极限过程。在这种一般情况下，正曲线空间(直到比例)在光滑黎曼流形上所起的作用与单位球面完全相同。我们对低维非负曲线空间的理解也在最近著名的Poincare猜想和几何化猜想的解决中发挥了关键作用。在更高的维度上，一般对具有非负或正曲率的流形或空间知之甚少。此外，只有少数结构和少量的例子为人所知。由于所有已知的例子都来自于群结构，并且具有相当大的对称群，这个建议的主要目的之一是通过描述或甚至可能分类具有大对称群的流形来扩展我们对具有正或非负曲率的流形的理解。这个结合了几何、拓扑和表示理论的程序已经获得了相当大的发展势头，并得到了一些分类结果，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、欧几里德空间和双曲空间正是具有常曲率和最大对称群特征的(单连通)空间。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)比常曲率空间中的“胖”。例如，如果一个空间中的测地线三角形比欧几里得平面(其中角度之和为180度)中的三角形“胖”，则该空间具有非负曲率。这种空间在几何中起着基本的作用，并形成了经典黎曼几何的推广，后者研究的是这类光滑正则空间。正、非负曲率，甚至“几乎非负”的曲率都起着特殊的作用，对它们的研究是必不可少的。就像在物理学的许多部分一样，我们在这个建议中的目的是分析并最终描述存在大群对称的正曲线空间和非负曲线空间(就像上面的经典常曲率模型空间一样)。这些研究还将为分析“几乎非正曲线空间”提供“模型”，从而为所有具有较低曲率界的空间的结构提供新的见解，并可能通过极限过程给出对具有非负曲率的流形的普遍追求的限制。

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.