Manifolds with Non-negative Curvature

AbstractAward：DMS-0504202主要研究者：Wolfgang Ziller主要研究者将继续对具有非负和正截面曲率的歧管进行研究。 在这个问题上，这已被广泛研究在早年的整体黎曼几何，只有少数障碍是已知的，其中主要是格罗莫夫的贝蒂数定理。另一方面，也有很少的已知的例子，主要是由一组等距的左不变度量所引起的。最近，在大量等距线存在下研究正曲流形取得了重大进展，部分原因是希望找到这类流形的新例子。最近我们得到了流形上正曲上齐性的局部分类，即群以一维商等距作用的流形。在这种分类中，人们留下了一个非常有趣的序列7-流形，令人惊讶的是，连接到自对偶爱因斯坦度量和3-Sasakian几何。我们计划研究这些流形是否具有正曲率的度量。对于非负弯曲的上齐一流形，有许多例子，由作者在以前的拨款建议，包括一些奇异的领域。另一方面，作者也指出奇异的Kervaire球不能携带这样的度量。这使得非负curvedmanifold的分类一个非常有趣的，虽然困难的问题，whichthe作者计划在未来的调查。 我们还计划研究几何和拓扑性质的已知例子流形与积极的，或更一般的非负截面曲率。作为下一步，我们计划研究具有低余齐性的正曲流形，由于常正曲率球面是最简单和最对称的黎曼流形，自然会问什么流形具有相似的几何性质，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可以导出什么样的全局结果，这是全局黎曼几何的一个主要领域。一个基本的未解决的问题是exoticspheres，即流形，看起来像球，但在其上的普通微积分是完全不同的，可以携带积极的curvedmetrics。 对称性是许多几何问题的一个重要方面，首席研究员计划在一个大的对称群的存在下研究具有正曲率或更一般的非负曲率的流形。这项调查的目标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.