论文标题
极地行动的主题
Topics in polar actions
论文作者
论文摘要
这些是2022年7月13日至15日，在德国斯图加特大学的几何学和拓扑研究所的一系列讲座的注释。我们假设对Riemannian歧管的等轴测动作进行基础知识，包括正常的Slice定理和主要的Orbit类型定理。第1节介绍了极地动作，并以海因兹，刘和奥尔莫斯的论点达到高潮，以表征它们在正常空间向主要轨道的分布的整合性方面。其他两个讲座专门用于Lytchak和Thorbergsson的两个结果。在第2节中，我们从度量的角度简要回顾了Riemannian Orbifolds，并解释了它们在适当和等距作用的轨道空间中的特征，从上面的切片表示的极性方面。在第3节中，我们介绍了他们的证据，证明了以下事实，即在非网状弯曲的歧管上以bott和samelson的意义上的变异完成作用是超极。附录包含对讲座中使用的某些结果的解释，即：Wilking的横向雅各比方程的一个或多或少的独立派生；讨论了卡坦和赫尔曼的标准，这些标准是完全测量的子曼群的存在，以及对对称空间的等距动作极性的标准。
These are the notes for a series of lectures at the Institute of Geometry and Topology of the University of Stuttgart, Germany, in July 13-15, 2022. We assume basic knowledge of isometric actions on Riemannian manifolds, including the normal slice theorem and the principal orbit type theorem. Lecture 1 introduces polar actions and culminates with Heintze, Liu and Olmos's argument to characterize them in terms of integrability of the distribution of normal spaces to the principal orbits. The other two lectures are devoted to two of Lytchak and Thorbergsson's results. In Lecture 2 we briefly review Riemannian orbifolds from the metric point of view, and explain their characterization of orbifold points in the orbit space of a proper and isometric action in terms of polarity of the slice representation above. In Lecture 3 we present their proof of the fact that variationally complete actions in the sense of Bott and Samelson on non-negatively curved manifolds are hyperpolar. The appendix contains explanations of some results used in the lectures, namely: a more or less self-contained derivation of Wilking's transversal Jacobi equation; a discussion of Cartan's and Hermann's criterions for the existence of totally geodesic submanifolds, and a criterion for the polarity of isometric actions on symmetric spaces.
