黎曼几何（英）（2016 春季学期）
中国科学技术大学数学科学学院
课程公告
(5/17) Lecture 18 uploaded.
(5/17) Midterm: May 28.
(5/21) Lecture 19 uploaded.
(5/24) Lecture 20 uploaded.
(5/24) PSet 3 uploaded. Due: June 20.
(5/30) Lecture 21 uploaded.
(6/7) Lecture 22 uploaded.
(6/7) Lecture 23 uploaded.
(6/11) Final topics uploaded.
(6/11) Lecture 24 uploaded.
(6/13) Lecture 25 uploaded.
(6/20) Lecture 26 uploaded.
(6/28) Lecture 27 uploaded.
课程信息
授课老师：
王作勤 (wangzuoq at ustc dot edu dot cn)
上课时间：
星期一上午 9：45am – 11：20am; 星期六晚上 19：30pm – 21：05pm
上课地点：
五教 5505
办公室：
管研楼1601
助教：
TBA ( at mail dot ustc dot edu dot cn)
答疑时间：
TBA
答疑地点：
TBA
参考书籍
M.do Carmo, Riemannian geometry
参考书籍
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry
参考书籍
Peter Petersen, Riemannian geometry
课程安排，讲义以及习题
课程安排可能会随着课程的进行而略有变动。
课程的讲义将在每堂课后上传。
| 序号 | 日期 | 内容 | 讲义 | 作业 |
|---|---|---|---|---|
| Lecture 1 | 02/29 | The Riemannian Metric | Lect 1 | PSet 1 : Due Mar. 26. |
| Lecture 2 | 03/05 | The Riemannian Distance | Lect 2 | |
| Lecture 3 | 03/07 | The Riemannian Measure | Lect 3 | |
| Lecture 4 | 03/12 | Linear Connections | Lect 4 | PSet 2 : Due April 25. |
| Lecture 5 | 03/14 | The Levi-Civita Connection | Lect 5 | |
| Lecture 6 | 03/19 | The Riemann Curvature Tensor | Lect 6 | |
| Lecture 7 | 03/21 | The Weyl Curvature Tensor | Lect 7 | |
| Lecture 8 | 03/26 | The Sectional and Ricci Curvature | Lect 8 | |
| Lecture 9 | 03/28 | More on Curvatures | Lect 9 | |
| Lecture 10 | 04/02 | The Parallel Transport | Lect 10 | PSet 3 : Due June 20. |
| Lecture 11 | 04/09 | The Geodesics | Lect 11 | |
| Lecture 12 | 04/11 | Variations and the Jacobi Fields | Lect 12 | |
| Lecture 13 | 04/25 | The Exponential Map | Lect 13 | |
| Lecture 14 | 04/30 | The Normal Coordinates | Lect 14 | |
| Lecture 15 | 05/07 | Geodesical Completeness and Convexity | Lect 15 | |
| Lecture 16 | 05/09 | Conjugate Points and Cut Points | Lect 16 | |
| Lecture 17 | 05/14 | The Index Form | Lect 17 | |
| Lecture 18 | 05/16 | The Theorems of Ambrose and Cartan-Hadamard | Lect 18 | Topics : Due July 2. |
| Lecture 19 | 05/21 | The Theorems of Bonnet-Meyer, Synge and Preissman | Lect 19 | |
| Lecture 20 | 05/23 | Rauch's Comparison Theore | Lect 20 | |
| 05/28 | Midterm Exam | |||
| Lecture 21 | 05/30 | The Hessian, Laplace and Toporogov Comparison Theorems | Lect 21 | |
| Lecture 22 | 06/04 | The Sphere Theorem | Lect 22 | |
| Lecture 23 | 06/06 | Critical Point Theory of Distance Function | Lect 23 | |
| Lecture 24 | 06/11 | The Volume Comparison Theorem and its Applications | Lect 24 | |
| Lecture 25 | 06/13 | The Hodge-Laplace Operator | Lect 25 | |
| Lecture 26 | 06/20 | The Hodge Theory | Lect 26 | |
| Lecture 27 | 06/25 | Bochner's Technique | Lect 27 |
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