Global Riemannian Geometry

AbstractAward：DMS-9971045首席研究员：Peter Petersen PI提出研究黎曼流形的几何和拓扑不变量之间的关系。本研究至少可分为两部分。有一般的有限性结果，其中一个简单地获得一些边界上的拓扑不变量从某些边界上的几何。PI提出了几个有限性现象的流形积分边界的曲率。本文还研究了三维流形的等谱紧性问题。有更多的具体情况下，人们可以计算非常具体的拓扑不变量从某些特定的几何信息。在这个标题下，人们还研究了各种捏现象，其中的目标是了解特定的拓扑类型的黎曼流形上的限制，它的一些几何不变量。PI提出研究非紧四维流形的Gauss-Bonnet型定理及其与欧氏量子引力研究的关系。这在紧致流形的情形下已经得到了很好的研究，但在非紧致流形上却很少被理解。本文还提出研究曲率和特征值的各种箍缩现象。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一方面几何量（如大小、体积和弯曲）与另一方面物体复杂性之间的关系。复杂性通常是通过物体上有多少个洞来衡量的，一个攀登架比一个甜甜圈更复杂，而甜甜圈又比一个球更复杂。 根据几何信息的好坏，人们可以获得复杂性的一般界限，但有时也可以完全确定对象是什么。例如，只有一个球具有这样的性质，即它在任何地方都以相同的方式弯曲。在几何量之间也有非常有趣的关系。例如，任何物体在任何地方都向同一个方向弯曲，其大小和体积都是有限的。这样的结果在爱因斯坦的广义相对论中也有对应的结果。这些结果关系到大爆炸、黑洞和宇宙最终命运的研究。PI研究的一个技术项目是弯曲测量不需要非常精确的可能性。换句话说，轻微视力受损的人对弯曲的想法和视力正常的人一样好。

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.