An Introduction to Riemannian Geometry      Jose Natario

第一章 可微分流形      簡介      [ExerciseDG101]

1. Topological Manifold
2. Differentiable Manifolds    Differentiable Maps
3. (1)Lie derivative (2)holonomy
   (1)Jacobi fields
   (2) Why are Killing fields relevant in physics ? [Notes on Killing fields]( by Ivo Terek)
   
4. metrics
5. Christoffel symbols   [Euler equation and Geodesics] by R.Herman
6. Divergence [在Riemannian manifold上的divergence]
7. immersion and embedding
8. Lie Groups
9. Frobenius可積分定理 [積分因子 Lie群的觀點] [可積的動力系統]
   [常均曲率曲面] ：H.C.Wente 1986 發展的技巧, 建立起常均曲率曲面與可積系統之關聯, 常均曲率環面因之而能被深入探討。
   如何理解三者的關係
10.  Geometric flows也稱為幾何演化方程
11. Tangent bundle
12. Manifolds with boundary

第二章 Differential Forms      [ExerciseDG201]

1. Differential Forms
2. Integration on Manifolds
3. Cartan magic formula
4. Stokes定理   [習作] 向量場的微積分 (1)散度 梯度 旋度
5. 曲面理論的活動標架法 (結構方程式) Gauss-Codazzi equations  Structure equations

    [Differential Forms and Connections] by Richard W.R. Darling

第三章 黎曼流形 (M,g)      [ExerciseDG301]

1. Affine connection   induced metric
   曲線長度在座標變換下不變
2. Covariant derivative , parallelism ,聯絡(connection)
3. Exponential map      normal coordinates(正則座標系)以簡化計算。
4. Geodesic    [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]
5. Hopf-Rinow定理
6. Einstein manifold

第四章 曲率      [ExerciseDG401]

1. (1)曲率(古典) (2)曲率 (3)S^2 IS^2 (傅科擺) (4)Ricci  (5)Cosmology(FLW model)  (6)Ricci Scalar for BH
2. [Geometry of the 3-phere] by Garret Sobczyk     [Ruth Gregory ResearchGate] GR and BH
3. 黎曼曲率張量
4. (Cartan) Structure equation (1)connection forms (2)curvature forms (3)R^n的結構方程
5. Gauss Bonnet定理
6. (1)sectional curvature (2)Ricci curvature (3)scalar curvature   [Ricci flow] (4)Ricci Scalar for the BH space
7. Isometric Immersions
8. Minimal surfaces
9. Hyperbolic plane

第五章 Geometric Mechanics      [ExerciseDG501]

第六章 Relativity                         [ExerciseDG601]

第七章  Manifold of constant curvature

第八章 能量的變分    Bonnet-Myers定理   Rauch比較定理  Morse 指標定理  Synge-Weistein定理

第九章 負曲率流形的基本群     complete manifolds    Hadamard定理