Lie Group Actions in Geometry

主要研究者Wolfgang Ziller计划继续研究李群在几何中出现的各种方式。特别强调的是上齐性单流形的几何和拓扑，即李群以一维商作用于其上的流形。在我们以前的建议中，我们证明了每一个上齐性流形都有一个具有非负截面曲率的度量。一个特别有趣的例子是Kervaire球体，它在许多维度上都是奇异的。我们将尝试在这些Kervaire球面上构造具有非负曲率的度量。另一个目标是对所有具有正截面曲率的上齐一流形进行分类，希望能找到一些新的例子，在7维和9维中有很好的候选者。我们也将研究齐次爱因斯坦度规的变分性质。在我们以前的建议中，我们证明了部分Palais Smale条件的满足，这使得我们能够实现莫尔斯理论和Lusternik Schnirelmann理论。整体黎曼几何中一个主要的兴趣是研究曲率具有特殊性质的流形，特别是那些曲率具有固定符号的流形。正曲率和非负曲率流形的研究始于整体黎曼几何的诞生。这种类型的例子很少，新的似乎很难构建。我们的建议的一个主要目标是构建这样的已知的例子。特别感兴趣的是奇异球，这是流形，看起来像球，但普通微积分是完全不同的。这些物体是40年前由米尔诺发现的，从那时起，几何学家们就对找到它们的几何描述感兴趣，其中自然的局部不变量看起来像球体，即曲率是正的或非负的。它们是否有度量的正曲率实际上是其中一个主要的开放问题的subject.Another主题的主要兴趣在全球黎曼几何是存在的爱因斯坦度量，这有许多应用在物理特别是建设新的模型卡鲁扎克莱因理论。在这方面，齐次爱因斯坦度量已经被各种物理学家研究了很多。我们开发了一个通用的变分理论齐次爱因斯坦度量，使人们能够找到许多新的例子，而不必解决代数方程的爱因斯坦方程减少到齐次的情况下，这可能是相当困难或不可能明确解决。

DMS - 0203697 PI: Wolfgang Ziller ABSTRACTThe principal investigator plans to continue his work on variousways in which Lie groups arise in geometry. Special emphasis will be put ongeometry and topology of cohomogeneity one manifolds, i.e. manifolds on which aLie group acts with one dimensional quotient. In our previous proposal we conjectured that every cohomogeneity one manifold has a metric with non-negative sectional curvature. A particularly interesting case are the Kervaire spheres which are exotic in many dimensions. We will try to constructmetrics with nonnegative curvature on these Kervaire spheres. A further goal is to classify all cohomogeneity one manifolds with positive sectional curvaturein the hope of finding some new examples, with good candidates available in dimension 7 and 9. We will also study variational properties of homogeneous Einstein metrics. In our previous proposal we showed that a partial Palais Smale condition is satisfied which enables one do carry out Morse theory and Lusternik Schnirelmann theory.A subject of major interest in global Riemannian geometry is studying manifolds whose curvature have special properties, in particular those whose curvature has a fixed sign. Manifolds with positive curvature or nonnegative curvature have been studied since the beginning of global Riemannian geometry. Very few examples of this type are known and new ones seem to be difficult to construct. A main objective of our proposal is to construct such knew examples. Of particular interest are exotic spheres, which are manifolds that looklike spheres but on which ordinary calculus is quite different. These objectswere discovered 40 years ago by Milnor and ever since then geometers wereinterested in finding a geometric description of them where the natural localinvariants look like spheres, i.e. where the curvature is positive or non-negative. Whether they have metrics of positive curvature is in fact one of the major open problems in the subject.Another subject of major interest in global Riemannian geometry is the existence of Einstein metrics, which have many applications in physics in particular to building new models of Kaluza Klein theory. In this context homogeneous Einstein metrics have been studied a lot by various physicists. We develop a general variational theory for homogeneous Einstein metrics which enable one to find many new examples without having to solve the algebraic equations which the Einstein equations reduce to in the homogenous case and which can be quite difficult or impossible to solve explicitly.