黎曼幾何筆記    習作簿

  1. 黎曼幾何與幾何分析      Jurgen Jost
  2. 黎曼幾何與譜理論          Olivier Lablee
  3. 幾何力學                          Darryl D Holm
  4. 物理學家用的黎曼幾何  侯伯宇                         [規範與纖維]
  5. 廣義相對論                       Amol Sasane               [研究重力的好幫手(日新)]
  6. 黎曼猜想                           Hugh Montgomery
Topics :
  1. 3-sphere
  2. divergence theorem:推廣到manifold上   On the divergence theorem on manifold] (這裡提到Henstock-Kurzweil integral)
  3. Killing fields
  4. Jacobi fields
  5. Minimal Surface
  6. Surface of revolution
  7. Torus]  [Geodesics]
  8. Spinning Top   Euler-Lagrange equation(1)   用 Euler-Lagrange equation[R.Herman] 求geodesic equation and Christoffel symbols
  9. [Ricci---curvatrer flow solitons]
  10.  [Comparison theorem]

第一章 向量場

  1. 向量場 張量 [Vectorfields]   [complete vector fields]  [holonomy角] 旋量[Spinor]
  2. 奇點的標數把Gauss-Bonnet定理(切線轉角定理是特例)與Hopf-Poincare定理結合在一起
     [Stokes theorem與電磁學] [Stokes theorem on manifold]
  3. 在曲率的脈絡上 [Christoffel symbols]對廣義相對論提供了數學(計算)基礎
    用Euler-Lagrange equation[R.Herman]可方便求之
  4. 各種幾何流線[Flows] 

         [Topological manifolds]   [RG01 Jose Natario 第一章 習作]   

  1. (1)multivariate calculus (2)vector field (3)

         [黎曼幾何簡介] 3.4.3節 Schwarzschild metric

第二章 微分式(Differential forms)

  1. [Differential forms]把計算整合起來 例如[Stokes] [Structure equation]
  2. Check a formula of differential form dw(X^Y)=
  3. 何謂可積   [Frobenius theorem] 有兩種形式 (1)vector fields (2)differential forms
    微分方程的可積性]   [可積的動力系統]   [ Spinning top]   [sin-Gordon 方程]
     [Frobenius theorem01] by Richard Palais  [Frobenius02]   
  4. [Inner product]

           [RG01 Jose Natario 第二章 習作]

第三章 黎曼流形

  1. Prove X is a 2-dim manifold in R^3 (Implicit function theorem)   [證明O(n)是一differential manifold]
  2. Riemannian metric要滿足三條件
  3. (1) Levi-Civita connection 與metric相容 torsion free (2)自旋聯絡
  4. Immersion and Embedding   [Immersion Ex01]
  5.  [Cartan structure equation]  活動標架法  [Affine connection]
  6. Parallel transport a vector
  7. [Parallel and Foucault] 鐘擺運動的方向是幻象 [從傅科擺到平面波的拓撲] by Pierre Delplace  and Antoine Venaile
  8. Calculus on manifold (1)Lie derivative (2)covariant derivative (3)exterior derivative
    流形上有三個微分運算 Cartan magic formula把它們串起來  [elementary proof] by Oleg Zubelevich [ResearchGate]
    Why are Killing fields relevant in physics ? [Notes on Killing fields]( by Ivo Terek)
  9. Symmetries and Killing vector fields  [Ex01divK=0]  [Ex02]  [Ex03]  [Ex04]
  10. 等距同構 isometry
  11. Exponential map 指數映射 是一個局部的可微同胚(local diffeomorphism) 可以建構(1)正則鄰域 (2)Jacobi field
  12. Geodesics     [Torus01]  by Mark L. Irons    [The geodesics of a surface of revolution]
    Surface of revolution in R^3 is a trivial case of intrinsic rotatiolal surface by Seher Kaya and Rafael Lopez
    [虛功原理與Euler-lagrange方程式] [the curvature and geodesics on Torus]
    [蟲洞的測地線]   [the geodsic equations for metric of a charged BH] [Worm hole]
  13. Hopf-Rinow theorem
  14. (1)S^2  (2)IxS^2  (3)S^3  (4)Hyperbolic plane    [Ricci curvature of S^3 by Cartan structure equatuon]   [Ex S^2不可能是一個Lie group]
    Geometry of the 3-phere] by Garret Sobczyk     [Ruth Gregory ResearchGate] GR and BH
  15. immersion and embedding   Tangent bundle   Einstein manifold
  16. [Hyperbolic(1)] by  William Schult [Hyperbolic(2)] by  Aiden Sheckler   [Hyperbolic (3)] 不要忘了還有H^3

[002 Cartan formalism] 有(1) p.18 把旋轉曲面看成IxS 求得 K Ric and R (2) p.19 先算IxS^2 再會去算p.22 S^3 (3) p.23 H^3  p.35 H^3 by Ivo Terek Couto

  [RG01 Jose Natario 第三章  習作]  [KillingandJacobi]

  1. 黎曼流形(M,g)  度量[metric] 決定流形的性質   [induced metric]
  2. 曲率[Curvature]作用在物質上 顯現出重力 energy-momentum影響時空 創造出曲率
  3. 黎曼曲率張量
    (1)截面曲率(sectional curvature) (2)Ricci curvature (3)scalar curvature
  4. Ricci tensor]說反映非歐幾何中體積的扭曲
    (1)據稱Sir Halmiton在衝浪時想到Ricci flow
    (2)Cedric Villani流連在羊角麵包店時 思考著最優運輸
    (3)Ricci curvature for BH
  5. Gauss-Bonnet theorem   [Gauss Bonnet Chern theorem] by Yin Li
  6. 幾何與廣義相對論中的純量曲率   Richard(Rick) Schoen [專訪]  [Yamabe問題與正質量問題]
  7. [Spacetime and Geometry 第三章 Curvature] 第四章 [Gravitation] [習作 and Answers] Sean M. Carroll 1966- [ResearchGate]

         [RG01 Jose Natario 第四章 習作] [Curvature of IS^2] [Ricci curvature] [Isotropic] [Exam2020]

[RicciSoliton]里奇孤子   [Sine-Gordon equation]

第五章 幾何力學

第六章 重力場

第七章 變分法

  1. 弧長的第一二變分式   [Va002]
  2. 面積的變分    PDE習作
  3. 能量的變分    [Comparison theorem] Bonnet-Myers定理   Rauch比較定理
  4. Morse index    [CMC球面的Morse index]  [Pseudosphere的Morse與譜分析]
  5. Synge-Weistein定理

第八章 李群

 [Lie groups]  一個電腦資訊科學家的微分幾何與李群筆記  Jean Gallier  [Jean Gallier] 這裡有很多書

第九章 向量叢

  1. bundle chart of S^1

第十章  CMC曲面   [CMC曲面]

  1. An Introduction to Riemannian Geometry     Jose Natario
  2. Riemannian Geometry                                   Manfredo P. do Carmo
  3. Spacetime and Geometry                              Sean Carroll     [ProfoundPhysics] [Spacetime and Geometry(Exercises)]
  4. 物理學家用微分幾何 侯伯元 侯伯宇        檔名 DGforP
  5. Differential Geometry in Physics                   Gabriel Lugo  [ResearchGate]
  6. Geometry of Manifolds                                 Richard L.Bishop
  7. Lectures on Geometry of Manifolds              Liviu I. Nicolaescu
  8. 大域微分幾何                                             黃武雄
  9. 微分幾何講義                                             陳省身
  10. Differental Geometry                                    杜武亮
  11. An Introduction to Manifolds                       杜武亮
  12. Differential Forms and Connections]            Richard W.R. Darling
  13. Riemannian Geometry                                  Peter Petersen
  14. Differential Forms for Physics Students        William O. Straub
  15. A course in modern mathematical physics    Peter Szekeres
  16. Differential Geometry with Application to Mechanics and Physics        Yves Talpaert
  17. Mathematical physics : Classical Mechanics  Andreas Kmauf
  18. Differential forms with applications to the physical sciences          Harley Flanders
  19. A Course in Differential Geometry by Thierry Aubin (1942~2009) Yamabe problem
  20. 教科書/參考書書評   [Books]                    [Differential Geometry論壇]