基本介紹
- 書名：辛幾何講義
- 作者：[美]Shlomo Sternberg
- 原版名稱：Lectures on Symplectic Geometry
- 譯者：李逸
- ISBN：9787302294986
- 定價：49元
- 出版社：清華大學出版社
- 出版時間：2012年10月01日
內容簡介,圖書目錄,
內容簡介
本書是美國著名數學家Shlomo Sternberg於2010年在清華大學教授辛幾何的講義，分為兩個部分。
第一部分（第1章~第10章）介紹了辛群、辛範疇、辛流形和Kostant-Souriau定理等內容。
第二部分（第11章~第16章）分別討論了Marle常秩嵌入定理、環面作用的凸性定理、Hamiltonian線性化定理和極小偶對。
本書可供從事辛幾何和微分幾何相關領域研究的學者參考，也可作為高年級本科生和研究生的教材和參考書。
圖書目錄
第1章導論和背景知識······················································································1
1.1
一些歷史································································································1
1.1.1Hamilton····················································································1
1.1.2Jacobi·························································································2
1.1.3Lie······························································································3
1.1.4Cartan························································································4
1.2
線性辛幾何····························································································5
1.2.1
辛向量空間·················································································5
1.2.2
基本例子····················································································6
1.2.3
辛正交補····················································································6
1.2.4
幾類特殊的子空間······································································6
1.2.5
正則形式····················································································7
1.3
辛群·······································································································8
1.3.1
辛群····························································································8
1.3.2
二維辛群:Sp(2)=SL(2,.)··························································8
1.3.3
Gauss定理·················································································8
1.4
線性Hamilton理論··············································································10
1.4.1
Maxwell電動力學·····································································10
1.4.2
Fresnel光學···············································································10
1.4.3
幾何光學···················································································11
1.4.4
線性光學···················································································11
1.4.5
Gaussian光學···········································································11
1.4.6
Gaussian光學中的射線追蹤······················································12
1.4.7
Gaussian光學轉換成Sp(2)·······················································13
1.4.8
Snell定律··················································································13
1.4.9
折射的矩陣形式········································································14
1.
4.10常折射率介質中的射線····························································15
1.4.11
薄透鏡·····················································································15
1.4.12
薄透鏡的焦平面·······································································15
1.
4.13共軛平面和薄透鏡方程····························································16
1.4.14
望遠鏡·····················································································16
1.4.15
主平面·····················································································17
1.5
Gaussian光學中的Hamilton方法························································17
1.5.1
Gaussian光學中的Hamilton方法············································17
1.5.2
Hamilton想法···········································································19
1.5.3
光程···························································································20
1.
5.4光程的一個重要公式·································································20
1.
5.5光程公式的一個特殊情形··························································20
1.5.6
光程公式的證明········································································20
第2章辛群········································································································23
2.1
基礎知識回顧························································································23
2.1.1
辛向量空間················································································23
2.1.2
最簡單的例子············································································23
2.1.3
子空間的特殊情況·····································································24
2.1.4
辛子空間···················································································24
2.1.5
正則形式···················································································24
2.1.6
Lagrangian子空間的存在性······················································25
2.1.7
相容Hermitian結構··································································25
2.2
極分解的使用························································································26
2.
2.1線性代數中一些事實的回顧······················································26
2.
2.2非負自伴隨矩陣的平方根··························································26
2.2.3
極分解·······················································································27
2.
2.4辛幾何中極分解的使用······························································27
2.2.5
群Sp(V)是連通的·····································································28
2.2.6
Sp(V)的維數·············································································28
2.2.7
Lagrangian子空間構成的空間的維數·······································29
2.
3辛群的坐標描述····················································································29
2.
4辛矩陣的特徵值····················································································30
2.5
Sp(V)的Lie代數··················································································31
2.6
Sp(V)中元素的極分解··········································································31
2.6.1回到Sp(V)中元素的極分解的一個斷言上································33
2.7sp(V)的Cartan分解············································································34
2.8Sp(V)的緊子群·····················································································34
2.9Sp(V)的Gaussian生成元····································································34
2.9.1線性光學···················································································34
第3章線性辛範疇·····························································································39
3.1範疇理論·······························································································39
3.1.1範疇的定義················································································39
3.1.2函子···························································································40
3.1.3反變函子···················································································40
3.1.4態射···························································································41
3.1.5對合函子···················································································41
3.1.6對換函子···················································································41
3.2集合和關係···························································································42
3.2.1有限關係的範疇········································································42
3.2.2DX是恆等態射idX·····································································43
3.2.3結合法則···················································································43
3.3範疇化“點”························································································43
3.3.1FinRel中的“點”····································································44
3.3.2態射作用在“點”上·································································44
3.3.3回到FinRel範疇上···································································44
3.3.4FinRel上的轉置········································································46
3.4線性辛範疇···························································································46
3.4.1Γ2
Γ1空間·················································································47
3.4.2纖維乘積或正合方格·································································48
3.4.3轉置···························································································48
3.4.4投射α:Γ2
Γ1→Γ2°Γ1··································································48
3.4.5線性典範關係的核和像······························································49
3.4.6證明Γ2°Γ1是Lagrangian····························································50
3.4.7結合法則···················································································50
3.5LinSym範疇和辛群··············································································51
第4章辛向量空間的Lagrangian子空間和進一步的Hamilton方法·················53
4.1與有限個Lagrangian子空間橫截的Lagrangian子空間·······················53
4.1.1Lagrangian-Grassmanian空間··················································54
4.1.2
.(V,M)的參數化·······································································54
4.1.3
基描述·······················································································55
4.2
.(V)上的Sp(V)作用···········································································55
4.2.1
Sp(V)可遷地作用在.(V)的橫截對上·······································55
4.2.2
Sp(V)不可遷地作用在.(V)的橫截三元組上····························56
4.2.3
sgn(bL)的顯式計算····································································58
4.3
生成函式——Hamilton想法的一個簡單例子·······································60
4.3.1和M*橫截的子空間···································································61
