Nonpositive Curvature: Geometric And Analytic Aspects
Nonpositive Curvature: Geometric And Analytic Aspects
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DOI：
10.1007/978-3-0348-8918-6
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发表时间：
1997-05
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影响因子：
--
通讯作者：
J. Jost
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文献类型：
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作者：
J. Jost
本书收录了一门博士专题课程“Nachelvorlesung”的课堂讲稿。学生，在苏黎世联邦理工学院在冬季学期95/96。因此，这些笔记是根据组织口头陈述材料的要求安排的，难度和陈述的水平也根据苏黎世的观众进行了调整。本课程的目的是介绍一些几何和分析的概念，这些概念在推进我们对非正曲率空间的理解方面很有用。特别是在最近几年，人们已经意识到，它往往是有用的系统的理解，而不是限制注意黎曼流形，但考虑更一般的类度量空间的广义非正曲率。其基本思想是分离出一个性质，一方面可以用距离函数单独表述，另一方面是黎曼流形上的非正截面曲率的特征，然后将此性质作为定义非正曲率度量空间的公理。瓦尔德、亚历山德罗夫、布泽曼和其他人都提出过这样的结构，我们将在第二章中系统地探讨它们。我们的关注点和治疗方法往往与现有文献不同。在第一章中，我们考虑了几类非正曲率的黎曼流形的例子，并解释了如何在各种几何背景下利用曲率的非正性或负性的条件。
The present book contains the lecture notes from a" Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpos itive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of generalized nonpositive curvature. The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi tive curvature. Such constructions have been put forward by Wald, Alexandrov, Busemann, and others, and they will be systematically explored in Chapter 2. Our focus and treatment will often be different from the existing literature. In the first Chapter, we consider several classes of examples of Riemannian manifolds of nonpositive curvature, and we explain how conditions about nonpos itivity or negativity of curvature can be exploited in various geometric contexts.