Lie Group Actions in Geometry

PI: Wolfgang Ziller【摘要】首席研究员计划继续研究几何中李群产生的各种方式。重点讨论了同质一元流形的几何和拓扑，即aLie群作用于一维商的流形。在我们之前的建议中，我们推测每一个同质的流形都有一个非负截面曲率的度规。一个特别有趣的例子是克维尔球它在很多维度上都很奇特。我们将尝试在这些克维尔球上构造非负曲率的度规。进一步的目标是对所有具有正截面曲率的同质1流形进行分类，希望找到一些新的例子，在7维和9维中有很好的候选。我们也将研究齐次爱因斯坦度量的变分性质。在我们之前的建议中，我们证明了部分的Palais Smale条件是满足的，这使得人们能够实现Morse理论和Lusternik Schnirelmann理论。在整体黎曼几何中，研究曲率具有特殊性质的流形，特别是曲率具有固定符号的流形，是一个重要的研究课题。具有正曲率和非负曲率的流形从黎曼几何开始就被研究。这种类型的例子已知的很少，新的例子似乎很难构建。我们建议的一个主要目标是构建这样的已知例子。特别有趣的是奇异的球体，它们是看起来像球体的流形，但是普通的微积分在它们上面是完全不同的。这些物体是米尔诺在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，其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。它们是否有正曲率的度规实际上是这门学科中主要的开放问题之一。整体黎曼几何的另一个重要课题是爱因斯坦度量的存在性，它在物理学中有许多应用，特别是在建立卡鲁扎克莱因理论的新模型方面。在这种背景下，齐次爱因斯坦度规已经被不同的物理学家研究了很多。我们发展了齐次爱因斯坦度量的一般变分理论，使人们能够找到许多新的例子，而不必求解爱因斯坦方程在齐次情况下简化为的代数方程，这可能是相当困难或不可能明确求解的。

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.