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內容簡介

　　在微分幾何和拓撲學中，人們常常處理偏微分等式和不等式組，它們不管加上什麼邊界條件總有無窮多個解。在1950年代人們發現，這種類型的微分關係（即等式或不等式）的可解性常常可以化為一個純粹的具同倫論性質的問題。在此情形下人們說：相應的微分關係滿足h-原理。h-原理的兩個著名例子是：黎曼幾何中Nash-Kuiper的Cl-等度嵌入理論和微分拓撲中的Smale-Hirsch浸沒理論，它們後來被Gromov轉換為建立h-原理的強有力的一般方法。
　　作者介紹瞭^一原理的兩個主要證明方法：完整性近似和凸積分。除瞭幾個著名的例外，h-原理的大部分例子都可以用這裏的方法來處理。《美國數學會經典影音係列：h-原理引論（英文版）》還特彆強調瞭辛幾何和切觸幾何的應用。
　　Gromov的名著Partial Differential Relations是麵嚮專傢的關於h-原理的百科全書，而《美國數學會經典影音係列：h-原理引論（英文版）》則是第1本關於此理論及其應用的能被廣泛接受的論著。《美國數學會經典影音係列：h-原理引論（英文版）》是關於解偏微分等式和不等式幾何方法的一本很好的研究生教材。學習幾何、拓撲和分析的人都可從中深受裨益。

內頁插圖

目錄

Preface
Intrigue

Part 1 Holonomic Approximation
Chapter 1． Jets and Holonomy
§1．1． Maps and sections
§1．2． Coordinate definition ofjets
§1．3． Invariant definition ofjets
§1．4． The space X (1)
§1．5． Holonomic sections of the jet space X (r)
§1．6． Geometric representation of sections of X (r)
§1．7． Holonomic splitting
Chapter 2． Thom Transversality Theorem
§2．1． Generic properties and transversality
§2．2． Stratified sets and polyhedra
§2．3． Thom Transversality Theorem
Chapter 3． Holonomic Approximation
§3．1． Main theorem
§3．2． Holonomic approximation over a cube
§3．3． Fiberwise holonomic sections
§3．4． Inductive Lemma
§3．5． Proof of the Inductive Lemma
§3．6． Holonomic approximation over a cube
§3．7． Parametric case
Chapter 4． Applications
§4．1． Functions without critical points
§4．2． Smale's sphere eversion
§4．3． Open manifolds
§4．4． Approximate integration of tangential homotopies
§4．5． Directed embeddings of open manifolds
§4．6． Directed embeddings of closed manifolds
§4．7． Approximation of differential forms by closed forms

Part 2 Differential Relations and Gromov's h-Principle
Chapter 5． Differential Relations
§5．1． What is a differential relation?
§5．2． Open and closed differential relations
§5．3． Formal and genuine solutions of a differential relation
§5．4． Extension problem
§5．5． Approximate solutions to systems of differential equations
Chapter 6． Homotopy Principle
§6．1． Philosophy of the h-principle
§6．2． Different flavors of the h-principle
Chapter 7． Open Diff V-Invariant Differential Relations
§7．1． Diff V-invariant differential relations
§7．2． Local h-principle for open Diff V-invariant relations
Chapter 8． Applications to Closed Manifolds
§8．1． Microextension trick
§8．2． Smale-Hirsch h-principle
§8．3． Sections transversal to distribution

Part 3 The Homotopy Principle in Symplectic Geometry
Chapter 9． Symplectic and Contact Basics
§9．1． Linear symplectic and complex geometries
§9．2． Symplectic and complex manifolds
……

Part 4 Convex Integration
Bibliography
Index
美國數學會經典影音係列：h-原理引論（英文版） [Introduction to the h-Principle] 下載 mobi epub pdf txt 電子書

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