Geometry and Topology in the Presence of Lower Curvature Bounds

作为经典欧氏几何和球面几何的自然扩展，具有非负或正曲率的流形几何自整体黎曼几何开始以来一直扮演着核心角色。这种作用在过去几十年中才被放大，因为具有非负或正曲率的空间在相当普遍的背景下自然出现，包括极限过程。 在这种普遍性中，正弯曲空间（直到标度）扮演着与单位球面完全相同的角色来平滑黎曼流形。我们对低维非负弯曲空间的理解在最近著名的庞加莱和几何化几何的解决方案中也发挥了关键作用。在高维空间中，关于具有非负曲率或正曲率的流形或空间，我们所知相对较少。此外，只有少数结构和适度数量的例子是已知的。由于所有已知的例子都来自于群的构造，并且具有相当大的对称群，这个提议的主要目的之一是通过描述甚至可能对具有大对称群的流形进行分类来扩展我们对具有正曲率或非负曲率的流形的理解。这个结合了几何学、拓扑学和表示论的程序已经获得了相当大的动力，并导致了几个分类结果，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。和双曲空间是具有常曲率和极大对称群的（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）比常曲率空间更“胖”。例如，如果空间中的测地线三角形比欧几里得平面中的测地线三角形“胖”（其中角度之和为180度），则空间具有非负曲率。这样的空间在几何学中起着基础性的作用，并且形成了经典黎曼几何的扩展，它处理这种类型的光滑和正则空间。正曲率、非负曲率、甚至“几乎非负”曲率的曲率起着特殊的作用，它们的研究对所有这些曲率都是必不可少的。正如物理学的许多部分一样，我们在这个提议中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（就像上面的经典常曲率模型空间一样）。这些调查也将提供“模型”分析“几乎非正弯曲的空间”，从而提供新的见解，所有空间的结构与较低的曲率界，并可能产生一般长期追求的限制流形上的非负曲率通过极限过程。

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.