莫爾斯理論入門 [An Invitation to Morse Theory] 下載 mobi epub pdf 電子書 2025

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[美] 尼古萊斯庫 著

齣版社： 世界圖書齣版公司

ISBN：9787510027291

版次：1

商品編碼：10762448

包裝：平裝

外文名稱：An Invitation to Morse Theory

開本：24開

齣版時間：2010-09-01

用紙：膠版紙

頁數：241

正文語種：英文

內容簡介

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.

目錄

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.

評分☆☆☆☆☆

應該是比較適閤初學者的書，而且也不是很厚，值得一看。

評分☆☆☆☆☆

在適當的小範圍內聯結任意兩點的測地綫是最短綫，所以測地綫又稱為短程綫。

評分☆☆☆☆☆

3 度量幾何的測地綫

評分☆☆☆☆☆

2 微分幾何的測地綫

評分☆☆☆☆☆

2.1 唯一性及存在性

評分☆☆☆☆☆

好書

評分☆☆☆☆☆

2.1 唯一性及存在性

評分☆☆☆☆☆

應該是比較適閤初學者的書，而且也不是很厚，值得一看。

評分☆☆☆☆☆

應該是比較適閤初學者的書，而且也不是很厚，值得一看。