Geometry and Topology in the Presence of Lower Curvature Bounds

存在较低曲率界的几何和拓扑

• 批准号: 0706791
• 负责人: Karsten Grove
• 金额: $ 36.42万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706791&HistoricalAwards=false
• 关键词: Geometry; Topology; Presence; Lower; Curvature

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.

作为经典欧几里得和球形几何形状的自然巨大扩展，自全球Riemannian几何形状开始以来，具有非阴性或正曲率的流形的几何形状起着核心作用。由于在过去的几十年中，这种角色仅在过去的几十年中被放大，因为具有非负或正曲率的空间在相当一般的环境中自然出现，包括极限过程。 在这种通用性中，正面弯曲的空间（直至缩放）的作用与单位球的作用完全相同，以使Riemannian歧管平滑。我们对低维非弯曲空间的理解在最近的著名庞加罗和几何化猜想的解决方案中也起着关键作用。在较高的维度中，关于非阴性或正曲率的歧管或空间，相对较少知道。同样只有少数几个结构和适度的示例。这一事实是，所有已知的例子都来自群体结构，并且具有相当大的对称性，该提案的主要目的之一是通过描述或可能通过对具有较大对称群体的人进行分类或可能对具有积极或非负曲率的流形进行了解。该程序结合了几何，拓扑和表示理论已经获得了相当大的动力，并且在构建许多具有非负曲率的新歧管以及积极曲率的新候选者的构建中，构造了球体的新候选者。比这些空间更弯曲的空间以几何表征的特征是通过大地三角形（侧面长度最短的三角形）比在恒定曲率空间中“胖”的特性。例如，如果空间中的大地三角形比欧几里得平面（其中角度为180度），则空间具有非负曲率。这样的空间在几何形状中起着基本作用，并形成了经典的riemannian几何形状的扩展，该几何形状涉及这种类型的平滑和规则空间。积极的，非负曲率甚至“几乎非阴性”曲率起着特定作用，他们的研究对所有人都是必不可少的。在物理学的许多部分中，我们在该建议中的目的是分析并最终描述存在大量对称性的正面弯曲空间和非弯曲空间（就像上面的经典恒定曲率模型空间一样）。这些研究还将提供“模型”，以分析“几乎非裂开的弯曲空间”，从而为所有曲率结合较低的空间的结构提供了新的见解，并可能通过极限过程对具有非阴性曲率的流形产生一般性的限制。