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

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[美] 尼古萊斯庫 著

圖書標籤:

• 莫爾斯理論
• 拓撲學
• 微分幾何
• 數學
• 高等數學
• 邀請
• 入門
• 流形
• 臨界點
• 同調理論

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.

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3 度量幾何的測地綫

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Morse theory

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3 度量幾何的測地綫

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2.2 局部最短性

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微分拓撲學的一個重要分支。通常指兩部分內容：一是微分流形上可微函數的莫爾斯理論，即臨界點理論；二是變分問題的莫爾斯理論，即大範圍變分法。

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2 微分幾何的測地綫

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在微分拓撲中，莫爾斯理論的技術給齣瞭一個非常直接的分析一個流形的拓撲的方法，它是通過研究該流形上的可微函數達成。根據莫爾斯的基本見解，一個流形上的一個可微函數在典型的情況下，很直接的反映瞭該流形的拓撲。莫爾斯理論允許人們在流形上找到CW結構和柄分解，並得到關於它們的同調群的信息。在莫爾斯之前，凱萊和麥剋斯韋在製圖學的情況下發展瞭莫爾斯理論中的一些思想。莫爾斯最初將他的理論用於測地綫（路徑的能量函數的臨界點）。這些技術被拉烏爾·博特用於他的著名的博特周期性定理的證明中。微分拓撲是一個處理在微分流形上的可微函數的數學領域。很自然地，它是在研究微分方程理論的過程中被提齣來的。微分幾何是用微積分來研究幾何的學問。這些領域非常接近，在物理學，特彆在相對論方麵有許多的應用。它們閤在一起還建立瞭可從動力係統觀點直接研究的、可微流形的幾何理論。 測地綫又稱大地綫或短程綫，數學上可視作直綫在彎麯空間中的推廣；在有度規定義存在之時，測地綫可以定義為空間中兩點的局域最短路徑。測地綫(geodesic)的名字來自對於地球尺寸與形狀的大地測量學(geodesy)。

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老闆指定的書，不錯，紙質很好，內容很好，適閤研究生

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好書