莫爾斯理論入門 [An Invitation to Morse Theory] 下載 mobi epub pdf 電子書 2024

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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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2 微分幾何的測地綫

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2 微分幾何的測地綫

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的分支。它是H.M.莫爾斯在20世紀30年代創立的。由莫爾斯理論得知 ，微分流形與其上的光滑函數緊密相關，利用光滑函數不僅能研究微分流形的局部性質，而且某些光滑函數例如莫爾斯函數包含瞭刻劃流形整體性質的豐富信息。莫爾斯理論主要分兩部分，一是臨界點理論，一是它在大範圍變分問題上的應用。一個莫爾斯函數也是一個非簡諧振子的一種錶達法。

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如果兩麯麵沿一麯綫相切，並且此麯綫是其中一個麯麵的測地綫，那麼它也是另一個麯麵的測地綫。

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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。

評分☆☆☆☆☆

評分☆☆☆☆☆

的分支。它是H.M.莫爾斯在20世紀30年代創立的。由莫爾斯理論得知 ，微分流形與其上的光滑函數緊密相關，利用光滑函數不僅能研究微分流形的局部性質，而且某些光滑函數例如莫爾斯函數包含瞭刻劃流形整體性質的豐富信息。莫爾斯理論主要分兩部分，一是臨界點理論，一是它在大範圍變分問題上的應用。一個莫爾斯函數也是一個非簡諧振子的一種錶達法。

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數學類基礎用書，值得參考

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