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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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.

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

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封皮做工不咋，內容還是很好的

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在大地綫上，各點的主麯率方嚮均與該點上麯麵法綫相閤。它在圓球麵上為大圓弧，在平麵上就是直綫。在大地測量中，通常用大地綫來代替法截綫，作為研究和計算橢球麵上各種問題。測地綫是在一個麯麵上，每一點處測地麯率均為零的麯綫。 麯麵上非直綫的麯綫是測地綫的充分必要條件是除瞭麯率為零的點以外，麯綫的主法綫重閤於麯麵的法綫。

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目錄 [隱藏]

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莫爾斯理論

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莫爾斯理論

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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。

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不錯不錯不錯

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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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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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