Lie Group Actions in Geometry

DMS-0203697 PI：沃尔夫冈·齐勒：首席研究员计划继续他的工作，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Alie群作用于其上的一维商的流形。在我们先前的建议中，我们猜想每一个上齐性流形都有一个截曲率非负的度量。一个特别有趣的例子是科瓦伊尔球体，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同调一流形进行分类，以期找到一些新的例子，在7维和9维都有很好的候选者。我们还将研究齐次爱因斯坦度量的变分性质。在我们以前的建议中，我们证明了部分Palais Smear条件是满足的，这使得我们能够实现Morse理论和Lusternik Schnirelmann理论。整体黎曼几何中的一个主要感兴趣的主题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。自整体黎曼几何诞生以来，具有正曲率或非负曲率的流形一直被研究。这种类型的例子很少为人所知，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。特别令人感兴趣的是奇异球体，这是一种看起来像球体但普通微积分截然不同的流形。这些天体是40年前由米尔诺发现的，从那时起，几何学家们一直对寻找它们的几何描述感兴趣，在那里，自然的定域不变量看起来像球体，即曲率为正或非负的地方。事实上，它们是否具有正曲率的度量是该学科的主要开放问题之一。全球黎曼几何中另一个主要感兴趣的主题是爱因斯坦度量的存在，它在物理中有许多应用，特别是在建立Kaluza Klein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规的一般变分理论，它使人们能够找到许多新的例子，而不必求解代数方程，在齐次情况下，爱因斯坦方程简化为代数方程，并且可能非常困难或不可能显式求解。

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.