Manifolds with Non-negative Curvature

摘要奖：DMS-0504202主要研究人员：沃尔夫冈·齐勒主要研究具有非负和正截面曲率的流形。在这个早年被广泛研究的课题中，只有很少的障碍是已知的，其中最主要的是格罗莫夫的Betti数定理。另一方面，已知的例子也很少，主要是由一组等距线取左不变度量的商而产生的。最近，在大量等距流形下研究正曲线流形已经取得了重大进展，部分原因是希望为这类新的例子找到好的候选者。最近，我们在流形上得到了正曲余齐性的局部分类，即流形上一个群以一维商等距作用的流形。在这个分类中，留下了一个非常有趣的7-流形序列，令人惊讶的是，它们与自对偶爱因斯坦度规和3-Sasakian几何相连。我们计划研究这些流形是否带有正曲率的度规。对于非负曲线上齐性流形，有许多例子，作者在以前的授权建议中构造了这些例子，包括一些关于奇异球面的例子。另一方面，作者还证明了奇异的Kervaire球面不能携带这样的度规。这使得非负曲线流形的分类成为一个非常有趣而又困难的问题，作者计划在未来进行研究。我们还计划研究具有正截面曲率或更一般地具有非负截面曲率的已知流形的几何和拓扑性质。作为下一步，我们计划研究具有低上齐性的正弯曲流形，由于常正曲率的圆形球面是最简单和最对称的黎曼流形，自然会问哪些流形具有相似几何性质的度量，即具有正曲率的度量。这符合一个自然的问题，即在局部几何假设下，人们可以得出什么样的全球后果，局部几何假设是全球黎曼几何的一个主要领域。一个基本的悬而未决的问题是，奇异球体，即看起来像球体但在其上与普通微积分有很大不同的流形，是否可以携带正曲线度量。对称性是许多几何问题的一个重要方面，主要研究者计划在一个大的对称群的存在下研究具有正的或更一般的非负曲率的流形。这项调查的目的之一是寻找新的例子。

AbstractAward: DMS-0504202Principal Investigator: Wolfgang ZillerThe principal investigator will continue his work on manifoldswith non-negative and positive sectional curvature. In thissubject, which has been studied extensively in the early years ofglobal Riemannian geometry, only few obstructions are known, themajor one being Gromov's Betti number theorem. On the other handthere are also very few known examples, mainly arising by takingquotients of left invariant metrics by a group ofisometries. Recently significant progress has been made bystudying positively curved manifolds under the presence of alarge group of isometries, partly in the hope of finding goodcandidates for new examples of this type. We recently obtained apartial classification of positively curved cohomogeneity onemanifolds, i.e. manifolds where a group acts isometrically withone dimensional quotient. In this classification one is left witha very interesting sequence of 7-manifolds which are,surprisingly, connected to self dual Einstein metrics and3-Sasakian geometry. We plan to investigate whether thesemanifolds carry a metric with positive curvature. Fornon-negatively curved cohomogeneity one manifolds there are manyexamples, constructed by the author in previous grant proposals,including some on exotic spheres. On the other hand the authoralso showed that the exotic Kervaire spheres cannot carry suchmetrics. This makes a classification of non-negatively curvedmanifolds a very interesting although difficult question, whichthe authors plans to investigate in the future. We also plan tostudy geometric and topological properties of the known examplesof manifolds with positive, or more generally non-negativesectional curvature. As a next step, we plan to study positivelycurved manifolds with low cohomogeneity.Since the round sphere of constant positive curvature is thesimplest and most symmetric Riemannian manifold, it is natural toask what manifolds carry metrics with similar geometricproperties, i.e. metrics with positive curvature. This fits intothe natural question of what global consequences one can deriveunder local geometric assumptions, a major area of globalRiemannian geometry. A basic unsolved question is whether exoticspheres, i.e. manifolds that look like spheres but on whichordinary calculus is quite different, can carry positively curvedmetrics. Symmetries are an important aspect of many geometricquestions and the principal investigator plans to study manifoldswith positive or more generally non-negative curvature under thepresence of a large symmetry group. One of the goals of thisinvestigation is the search for new examples.