Global Riemannian Geometry

摘要/ abstract摘要：dms -9971045项目负责人：Peter petersenpi提出研究黎曼流形的几何不变量和拓扑不变量之间的关系。这项研究至少可以分为两个部分。有一般有限的结果，可以简单地从几何上的某些界得到某些界本体不变量。提出了几种具有曲率积分界的流形的有限现象。本文还研究了3流形的等谱紧性问题。在更特殊的情况下，人们可以从特定的几何信息中计算出非常特定的拓扑不变量。在这个标题下，人们还研究各种捏缩现象，其目标是了解黎曼流形的特定拓扑类型及其某些几何不变量的限制。提出研究非紧4流形的Gauss-Bonnet型定理及其与欧几里得量子引力研究的关系。这在紧流形的情况下得到了很好的研究，但对于非紧流形却知之甚少。本文还提出研究曲率和特征值的各种捏缩现象。最后，PI将对Gromoll-Meyer球进行研究，并试图确定它是否允许具有正截面曲率的度量。PI研究的主要目标是理解几何量（如尺寸、体积和弯曲）和物体复杂性之间的关系。复杂性通常是通过物体有多少洞来衡量的，攀爬架比甜甜圈更复杂，而甜甜圈又比球更复杂。根据几何信息的好坏，我们可以获得复杂性的一般界限，但有时也可能完全确定对象是什么。例如，只有球具有在任何地方都以同样的方式弯曲的特性。几何量本身之间也有很有趣的关系。例如，任何向同一方向弯曲的物体在大小和体积上都是有限的。这样的结果在爱因斯坦的广义相对论中也有对应的结果。这些结果与宇宙大爆炸、黑洞和宇宙最终命运的研究有关。PI研究的一项技术项目是一种不需要非常精确的弯曲测量的可能性。换句话说，轻度视力受损的人对弯曲的看法与视力良好的人一样好。

AbstractAward: DMS-9971045Principal Investigator: Peter PetersenThe PI proposes to investigate the relationship between geometricand topological invariants of Riemannian manifolds. The study canbe divided up into at least two parts. There are generalfiniteness results where one simply obtains some bounds ontopological invariants from certain bounds on the geometry. ThePI proposes to work on several finiteness phenomena for manifoldswith integral bounds for the curvature. The isospectralcompactness problem for 3-manifolds will also beinvestigated. There are more specific situations where one cancompute very specific topological invariants from certainspecific geometric information. Under this heading one alsostudies various pinching phenomena, where the goal is tounderstand the specific topological type of a Riemannian manifoldwith restrictions on some of its geometric invariants. The PIproposes to study Gauss-Bonnet type theorems for noncompact4-manifolds and their relationship with the study of Euclideanquantum gravity. This is well studied in the setting of compactmanifolds, but much less understood for noncompact manifolds. Itis also proposed to investigate various pinching phenomena bothfor curvature and eigenvalues. Finally the PI will undertake astudy of the Gromoll-Meyer sphere and try to determine whether itadmits metrics with positive sectional curvature.The main goal of the PI's research centers on understanding therelationship between on one hand geometric quantities such assize, volume and bending and on the other hand the complexity ofthe objects. Complexity is often measured by how many holes theobject has, a jungle gym is more complex than a doughnut which inturn is more complex than a ball. Depending on how good onesgeometric information is one can obtain very general bounds onthe complexity, but it is also possible to sometimes completelydetermine what the object is. For instance, only a ball has theproperty that it bends in the same manner everywhere. There arealso very interesting relationships between the geometricquantities themselves. For instance, any object that bends in thesame direction everywhere is bounded in size and volume. Suchresults also have counterparts in Einstein's theory of generalrelativity. There such results are related to the study of theBig Bang, black holes, and the ultimate fate on the universe. Onetechnical item the PI studies is the possibility that onesmeasurement of bending does not have to be very precise. In otherwords, slightly vision impaired people's ideas of bending are asgood as those of people with perfect eye sight.