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辛几何讲义(Lectures on Symplectic Geometry)
《辛几何讲义(Lectures on Symplectic Geometry)》是2012年清华大学出版社出版的图书，作者[美]Shlomo Sternberg 着 李 逸 编译。
基本介绍
- 书名：辛几何讲义
- 作者：[美]Shlomo Sternberg
- 原版名称：Lectures on Symplectic Geometry
- 译者：李逸
- ISBN：9787302294986
- 定价：49元
- 出版社：清华大学出版社
- 出版时间：2012年10月01日
内容简介
本书是美国着名数学家Shlomo Sternberg于2010年在清华大学教授辛几何的讲义，分为两个部分。
第一部分（第1章~第10章）介绍了辛群、辛範畴、辛流形和Kostant-Souriau定理等内容。
第二部分（第11章~第16章）分别讨论了Marle常秩嵌入定理、环面作用的凸性定理、Hamiltonian线性化定理和极小偶对。
本书可供从事辛几何和微分几何相关领域研究的学者参考，也可作为高年级本科生和研究生的教材和参考书。
图书目录
第1章导论和背景知识······················································································1
1.1
一些历史································································································1
1.1.1Hamilton····················································································1
1.1.2Jacobi·························································································2
1.1.3Lie······························································································3
1.1.4Cartan························································································4
1.2
线性辛几何····························································································5
1.2.1
辛向量空间·················································································5
1.2.2
基本例子····················································································6
1.2.3
辛正交补····················································································6
1.2.4
几类特殊的子空间······································································6
1.2.5
正则形式····················································································7
1.3
辛群·······································································································8
1.3.1
辛群····························································································8
1.3.2
二维辛群:Sp(2)=SL(2,.)··························································8
1.3.3
Gauss定理·················································································8
1.4
线性Hamilton理论··············································································10
1.4.1
Maxwell电动力学·····································································10
1.4.2
Fresnel光学···············································································10
1.4.3
几何光学···················································································11
1.4.4
线性光学···················································································11
1.4.5
Gaussian光学···········································································11
1.4.6
Gaussian光学中的射线追蹤······················································12
1.4.7
Gaussian光学转换成Sp(2)·······················································13
1.4.8
Snell定律··················································································13
1.4.9
折射的矩阵形式········································································14
1.
4.10常折射率介质中的射线····························································15
1.4.11
薄透镜·····················································································15
1.4.12
薄透镜的焦平面·······································································15
1.
4.13共轭平面和薄透镜方程····························································16
1.4.14
望远镜·····················································································16
1.4.15
主平面·····················································································17
1.5
Gaussian光学中的Hamilton方法························································17
1.5.1
Gaussian光学中的Hamilton方法············································17
1.5.2
Hamilton想法···········································································19
1.5.3
光程···························································································20
1.
5.4光程的一个重要公式·································································20
1.
5.5光程公式的一个特殊情形··························································20
1.5.6
光程公式的证明········································································20
第2章辛群········································································································23
2.1
基础知识回顾························································································23
2.1.1
辛向量空间················································································23
2.1.2
最简单的例子············································································23
2.1.3
子空间的特殊情况·····································································24
2.1.4
辛子空间···················································································24
2.1.5
正则形式···················································································24
2.1.6
Lagrangian子空间的存在性······················································25
2.1.7
相容Hermitian结构··································································25
2.2
极分解的使用························································································26
2.
2.1线性代数中一些事实的回顾······················································26
2.
2.2非负自伴随矩阵的平方根··························································26
2.2.3
极分解·······················································································27
2.
2.4辛几何中极分解的使用······························································27
2.2.5
群Sp(V)是连通的·····································································28
2.2.6
Sp(V)的维数·············································································28
2.2.7
Lagrangian子空间构成的空间的维数·······································29
2.
3辛群的坐标描述····················································································29
2.
4辛矩阵的特徵值····················································································30
2.5
Sp(V)的Lie代数··················································································31
2.6
Sp(V)中元素的极分解··········································································31
2.6.1回到Sp(V)中元素的极分解的一个断言上································33
2.7sp(V)的Cartan分解············································································34
2.8Sp(V)的紧子群·····················································································34
2.9Sp(V)的Gaussian生成元····································································34
2.9.1线性光学···················································································34
第3章线性辛範畴·····························································································39
3.1範畴理论·······························································································39
3.1.1範畴的定义················································································39
3.1.2函子···························································································40
3.1.3反变函子···················································································40
3.1.4态射···························································································41
3.1.5对合函子···················································································41
3.1.6对换函子···················································································41
3.2集合和关係···························································································42
3.2.1有限关係的範畴········································································42
3.2.2DX是恆等态射idX·····································································43
3.2.3结合法则···················································································43
3.3範畴化“点”························································································43
3.3.1FinRel中的“点”····································································44
3.3.2态射作用在“点”上·································································44
3.3.3回到FinRel範畴上···································································44
3.3.4FinRel上的转置········································································46
3.4线性辛範畴···························································································46
3.4.1Γ2
Γ1空间·················································································47
3.4.2纤维乘积或正合方格·································································48
3.4.3转置···························································································48
3.4.4投射α:Γ2
Γ1→Γ2°Γ1··································································48
3.4.5线性典範关係的核和像······························································49
3.4.6证明Γ2°Γ1是Lagrangian····························································50
3.4.7结合法则···················································································50
3.5LinSym範畴和辛群··············································································51
第4章辛向量空间的Lagrangian子空间和进一步的Hamilton方法·················53
4.1与有限个Lagrangian子空间横截的Lagrangian子空间·······················53
4.1.1Lagrangian-Grassmanian空间··················································54
4.1.2
.(V,M)的参数化·······································································54
4.1.3
基描述·······················································································55
4.2
.(V)上的Sp(V)作用···········································································55
4.2.1
Sp(V)可迁地作用在.(V)的横截对上·······································55
4.2.2
Sp(V)不可迁地作用在.(V)的横截三元组上····························56
4.2.3
sgn(bL)的显式计算····································································58
4.3
生成函式——Hamilton想法的一个简单例子·······································60
4.3.1和M*横截的子空间···································································61