黎曼几何（英）（2024 春季学期）
中国科学技术大学数学科学学院
课程公告
(02/22) Welcome! This course will be taught in English.
(02/27) Lecture 1 uploaded.
(03/05) Lecture 2 and Lecture 3 uploaded.
(03/14) Lecture 4 and Lecture 5 uploaded.
(03/24) Lecture 6 and Lecture 7 uploaded.
(03/24) PSet 1 uploaded. Due April 09.
(03/31) Lecture 8 and Lecture 9 uploaded.
(04/08) Lecture 10 and Lecture 11 uploaded.
(04/15) PSet 2 uploaded. Due April 28.
(04/15) Midterm：April 28, in class.
(04/20) Lecture 12, 13, 14, 15 uploaded.
(05/07) Lecture 16, 17, 18 uploaded.
(05/07) PSet 3 uploaded. Due May 16.
(05/16) Lecture 19, 20 uploaded.
(05/26) PSet 4 uploaded. Due June 4.
(05/29) Topics for A+ uploaded. Due June 20.
(06/03) Lecture 21, 22, 23, 24, 25 uploaded.
(06/11) Lecture 26, 27, 28, 29 uploaded.
课程信息
授课老师：
王作勤 (wangzuoq at ustc dot edu dot cn)
上课时间：
星期二下午 14：00pm – 15：35pm; 星期四晚上 19：30pm – 21：55pm
上课地点：
五教 5302
办公室：
管研楼1601
助教：
叶星辰 (yexkif_1oclock at mail dot ustc dot edu dot cn)
助教：
李禹龙 (liyulon at mail dot ustc dot edu dot cn)
答疑时间：
周日下午 14:00 - 17:00 pm
答疑地点：
5302
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
参考书籍
T. Sakai, 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。
A+大作业 Topics 。
| 序号 | 日期 | 内容 | 讲义 | 作业 |
|---|---|---|---|---|
| Lecture 1 | 02/27 | Introduction | Lect 1 | |
| Lecture 2 | 02/29 | The Riemannian Metric | Lect 2 | PSet 1 : Due April 09. |
| Lecture 3 | 03/05 | The Riemannian Distance | Lect 3 | |
| Lecture 4 | 03/07 | The Riemannian Measure | Lect 4 | |
| Lecture 5 | 03/12 | Linear Connections | Lect 5 | |
| Lecture 6 | 03/14 | The Levi-Civita Connection | Lect 6 | |
| Lecture 7 | 03/19 | The curvature tensor | Lect 7 | PSet 2 : Due April 28. |
| Lecture 8 | 03/21 | The Riemann curvature tensor and its decomposition | Lect 8 | |
| Lecture 9 | 03/26 | The Sectional and Ricci Curvature | Lect 9 | |
| Lecture 10 | 03/28 | Spaces with constant curvature | Lect 10 | |
| Lecture 11 | 04/02 | The method of moving frame | Lect 11 | |
| Lecture 12 | 04/09 | Geodesics as self-parallel curves on manifolds with connection | Lect 12 | PSet 3 : Due May 16. |
| Lecture 13 | 04/11 | Geodesics as self-parallel curves on Riemannian manifolds | Lect 13 | |
| Lecture 14 | 04/16 | Existence of length-minimizing geodesics | Lect 14 | |
| Lecture 15 | 04/18 | Completeness: Rinow-Hopf theorem and Ambrose theorem | Lect 15 | |
| Lecture 16 | 04/23 | Variation formulae | Lect 16 | |
| 04/28 | Midterm Cover: Lecture 1-Lecture 15 | |||
| Lecture 17 | 04/25 | Jacobi fields | Lect 17 | PSet 4 : Due June 4. |
| Lecture 18 | 04/30 | First applications of Jacobi fields to curvature | Lect 18 | |
| Lecture 19 | 05/07 | Conjugate points and applications | Lect 19 | |
| Lecture 20 | 05/09 | The index form | Lect 20 | |
| Lecture 21 | 05/14 | Cut locus | Lect 21 | |
| Lecture 22 | 05/16 | Various theorem on curvature and topology | Lect 22 | |
| Lecture 23 | 05/21 | Rauch comparison theorem | Lect 23 | |
| Lecture 24 | 05/23 | The global Hessian and Toponogor comparison theorem | Lect 24 | |
| Lecture 25 | 05/28 | The Laplacian and volume comparison theorem | Lect 25 | |
| Lecture 26 | 05/30 | Applications of the volume comparison theorem | Lect 26 | |
| Lecture 27 | 06/04 | The sphere theorem | Lect 27 | |
| Lecture 28 | 06/06 | Bochner's technique | Lect 28 | |
| Lecture 29 | 06/11 | Eigenvalues of the Laplacian | Lect 29 | |
| 06/13 | Final Cover: Lecture 1-Lecture 29 |
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