h-原理引论（影印版）
￥99.00定价
作者： Y.Eliashberg，N.Mishachev
出版时间：2017-01
出版社：高等教育出版社
- 高等教育出版社
- 9787040469028
- 1版
- 74866
- 46253691-3
- 精装
- 16开
- 2017-01
- 320
- 206
- 理学
- 数学
- O186.1
- 数学类
- 研究生及以上
作者简介
内容简介
在微分几何和拓扑学中，人们常常处理微分方程组和偏微分不等式，它们不管加上什么边界条件总有无穷多个解。在1950年代人们发现，这种类型的微分关系（即等式或不等式）的可解性常常可以化为一个纯粹的具同伦论性质的问题。在此情形下人们说：相应的微分关系满足 h-原理。h-原理的两个著名例子是：黎曼几何中Nash-Kuiper的 C1-等度嵌入理论和微分拓扑中的Smale-Hirsch浸没理论，它们后来被Gromov转换为建立h-原理的强有力的一般方法。
作者介绍了h-原理的两个主要证明方法：完整性近似和凸积分。除了几个著名的例外，h-原理的大部分例子都可以用这里的方法来处理。本书还特别强调了辛几何和切触几何的应用。
作者的名著Partial Differential Relations是面向专家的关于h-原理的百科全书，而本书则是第一本关于此理论及其应用的能被广泛接受的论著。本书是关于解偏微分等式和不等式几何方法的一本很好的数学著作。学习几何、拓扑和分析的人都可从中深受裨益。
目录
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
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
