Lie Group Actions in Geometry

几何中的李群作用

• 批准号: 9971756
• 负责人: Wolfgang Ziller
• 金额: $ 19.47万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971756&HistoricalAwards=false
• 关键词: Lie; Group; Actions; Geometry

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.

项目编号：dms -9971756首席研究员：Wolfgang ziller首席研究员计划继续研究几何中李群产生的各种方式。重点将放在几何和拓扑的同质一流形，即。李群作用于一维商的流形。一个目标将是证明每一个同质单流形都有一个非负截面曲率的度规，这将意味着许多奇异球承认一个非负截面曲率的度规。第二个目标是对具有正截面曲率的同质1流形进行分类，希望找到一些具有正截面曲率的紧流形的新例子。对于非紧流形，我们将检验Cheeger Gromollsoul定理的逆定理，该定理要求紧非负弯曲流形上的哪些向量束允许具有非负曲率的完全度量。这可以通过检查对矢量束的主束的一个作用的同质性来完成。进一步的研究包括有限维李群的基本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，黎曼几何的一个重要课题是奇异球的几何，它是一种流形，看起来像球，但在其上的普通微积分却大不相同。这些天体是米尔诺在40年前发现的，从那时起，几何学家们就对寻找它们的几何描述很感兴趣，在这些几何描述中，自然的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，并有进一步的计划寻找更多这样的例子。一个自然的问题是，是否可以将它们变形为具有正曲率的度量，这是全局黎曼几何中最有趣的开放问题之一。我们的主要技术是检验这些物体和相关流形的对称性。大量的对称性通常意味着有趣的几何和拓扑性质，并且一直是许多数学学科的关键成分。我们能够构造许多具有非负曲率的新流形和如此大的对称群，这在几何和拓扑上都有许多有趣的应用。