莫尔斯理论入门 [An Invitation to Morse Theory] pdf epub mobi txt 电子书 下载 2025

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[美] 尼古莱斯库 著

出版社： 世界图书出版公司

ISBN：9787510027291

版次：1

商品编码：10762448

包装：平装

外文名称：An Invitation to Morse Theory

开本：24开

出版时间：2010-09-01

用纸：胶版纸

页数：241

正文语种：英文

内容简介

As the the title suggests， the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
Suppose M is a smooth， compact manifold， which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology， and we attempt a “slicing” technique.

目录

Preface
Notations and conventions
1 Morse Functions
1.1 The Local Structure of Morse Functions
1.2 Existence of Morse Functions

2 The Topology of Morse Functions
2.1 Surgery，Handle Attachment.and Cobordisms
2.2 The Topology of Sublevel Sets
2.3 Morse Inequalities
2.4 Morse-Smale Dynamics
2.5 Morse-Floer Homology
2.6 Morse-Bott Functions
2.7 Min-Max Theory

3 Applications
3.1 The Cohomology of Complex Grassmannians
3.2 Lefschetz Hyperplane Theorem
3.3 Symplectic Manifolds and Hamiltonian Flows
3.4 Morse Theory of Moment Maps
3.5 S1-Equivariant Localization

4 Basics of Comple X Morse Theory
4.1 Some Fundamental Constructions
4.2 Topological Applications of Lefschetz Pencils
4.3 The Hard Lefschetz Theorem
4.4 Vanishing Cycles and Local Monodromy
4.5 Proofofthe Picard Lefschetz formula
4.6 Global Picard-Lefschetz Formulae

5 Exercises and Solutions
5.1 Exercises
5.2 Solutions to Selected Exercises
References
Index

前言/序言

　　As the the title suggests， the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
　　Suppose M is a smooth， compact manifold， which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology， and we attempt a “slicing” technique.
　　We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersectM along a submanifold （slice）. The manifold can be recovered by continuouslystacking the slices on top of each other in the same order as they were cut out of M.
　　Think of the collection of slices as a deck of cards of various shapes. If welet these slices continuously pile up in the order they were produced， we noticean increasing stack of slices. As this stack grows， we observe that there aremoments of time when its shape suffers a qualitative change. Morse theoryis about extracting quantifiable information by studying the evolution of theshape of this growing stack of slices.

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小册子，参考

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在适当的小范围内联结任意两点的测地线是最短线，所以测地线又称为短程线。

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在适当的小范围内联结任意两点的测地线是最短线，所以测地线又称为短程线。

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数学类基础用书，值得参考

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老板指定的书，不错，纸质很好，内容很好，适合研究生

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3 度量几何的测地线

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的分支。它是H.M.莫尔斯在20世纪30年代创立的。由莫尔斯理论得知 ，微分流形与其上的光滑函数紧密相关，利用光滑函数不仅能研究微分流形的局部性质，而且某些光滑函数例如莫尔斯函数包含了刻划流形整体性质的丰富信息。莫尔斯理论主要分两部分，一是临界点理论，一是它在大范围变分问题上的应用。一个莫尔斯函数也是一个非简谐振子的一种表达法。

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在大地线上，各点的主曲率方向均与该点上曲面法线相合。它在圆球面上为大圆弧，在平面上就是直线。在大地测量中，通常用大地线来代替法截线，作为研究和计算椭球面上各种问题。测地线是在一个曲面上，每一点处测地曲率均为零的曲线。 曲面上非直线的曲线是测地线的充分必要条件是除了曲率为零的点以外，曲线的主法线重合于曲面的法线。

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在适当的小范围内联结任意两点的测地线是最短线，所以测地线又称为短程线。