Lie Group Actions in Geometry

摘要奖：DMS-9971756主要研究员：沃尔夫冈·齐勒主要研究员计划继续研究几何中李群产生的各种方式。我们将特别强调上齐性流形的几何和拓扑，即李群作用于一维商的流形。一个目的是证明流形上的每个上齐次流形都有一个截面曲率非负的度量，这意味着许多奇异球体都有一个截面曲率非负的度量。第二个目标是对具有正截面曲率的同调一流形进行分类，希望找到一些具有正截面曲率的紧致流形的新例子。对于非紧流形，我们将研究Cheeger GromollSoul定理的逆，该定理要求紧非负曲线流形上的哪个向量丛允许具有非负曲率的完备度量。这可以通过检查向量束的主丛上的同质性一个动作来完成。进一步的研究包括有限维李群的原子群的分类和齐次空间上标量曲率泛函的整体变分性质。在过去的30年里，黎曼几何的一个主要感兴趣的主题一直是奇异球面的几何，它是看起来像球面的流形，但普通计算在它上面有很大的不同。这些天体是40年前由米尔纳发现的，从那时起，几何学家们一直在寻找对它们的年龄统计描述，在这些描述中，自然局部不变量看起来像球体，即曲率为正或非负的地方。在我们的项目中，我们发现了许多曲率非负的奇异球体的新例子，并有进一步的计划找到更多这样的例子。一个自然的问题是，人们是否可以将这些变形为具有正曲率的度量，这是全球黎曼测量学中最有趣的公开问题之一。我们的主要技术是研究这类对象及其相关流形的对称性。一大组对称通常隐含着有趣的几何和拓扑性质，并且一直是许多数学科目的关键要素。我们能够构造出许多具有非负曲率和如此大的对称群的新流形，这具有许多有趣的应用，包括几何和拓扑学。

AbstractAward: DMS-9971756Principal Investigator: Wolfgang ZillerThe principal investigator plans to continue his work on variousways in which Lie groups arise in geometry. Special emphasis willbe put on geometry and topology of cohomogeneity one manifolds,i.e. manifolds on which a Lie group acts with one dimensionalquotient. One goal will be to show that every cohomogeneity onemanifold has a metric with non-negative sectional curvature,which would imply that many exotic spheres admit a metric withnon-negative sectional curvature. The second goal will be toclassify cohomogeneity one manifolds with positive sectionalcurvature, with the hope of finding some new examples of compactmanifolds with positive sectional curvature. For non-compactmanifolds, we will examine the converse to the Cheeger Gromollsoul theorem, which asks which vector bundles over compactnon-negatively curved manifolds admit complete metrics withnon-negative curvature. This can be done by examiningcohomogeneity one action on the principal bundle of the vectorbundle. Further studies include the classification of primitivesubgroups of finite dimensional Lie groups and global variationalproperties of the scalar curvature functional on homogeneousspaces.A subject of major interest in Riemannian geometry over the last30 years has been the geometry of exotic spheres, which aremanifolds that look like spheres but on which ordinary calculusis quite different. These objects were discovered 40 years ago byMilnor and ever since then geometers were interested in finding ageometric description of them where the natural local invariantslook like spheres, i.e. where the curvature is positive ornon-negative. In our project we have found many new examples ofexotic spheres where the curvature is non-negative and havefurther plans for finding more such examples. A natural questionis if one can deform these to metrics with positive curvature,one of the most intriguing open problems in global Riemanniangeometry. Our major technique is to examining symmetryproperties of such objects and related manifolds. A large groupof symmetries usually implies interesting geometric andtopological properties and has always been a key ingredient inmany mathematical subjects. We were able to construct many newmanifolds with non-negative curvature and such large symmetrygroups, which has a number of interesting applications, both ingeometry and topology.