National Taiwan Normal University Course Outline
 Spring , 2023

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I.Course information
Serial No. 2636 Course Level Undergraduate / Master / PhD
Course Code MAC7002 Chinese Course Name 黎曼幾何（二）
Course Name Riemannian Geometry (II)
Department Department of Mathematics
Two/one semester 1 Req. / Sel. Sel.
Credits 3.0 Lecturing hours Lecture hours: 3
Prerequisite Course
Comment
Course Description
Time / Location Tue. 2-4 Gongguan MA3-10
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Introduction of the advanced knowledge in Riemannian geometry College:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　1-3 Being able to think mathematically and critically
　1-4 Possessing the abilities to propose and solve questions in advanced mathematics
　1-5 Being able to use mathematics as tools to learn other subjects
Master:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　1-3 Being able to think mathematically and critically
　1-4 Possessing the abilities to propose and solve questions in advanced mathematics
　1-5 Being able to use mathematics as tools to learn other subjects
Doctor:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　1-3 Being able to think mathematically and critically
　1-4 Possessing the abilities to propose and solve questions in advanced mathematics
　1-5 Being able to use mathematics as tools to learn other subjects

II. General Syllabus
Instructor(s) LIN, Chun-Chi/ 林俊吉
Schedule

6. Some Background Knowledge and Review of Riemannian Geometry  (3-4 weeks)
 
6.1. Tangent vectors and Lie derivatives;   
6.2. Frobenius Theorem;
6.3. A review on Riemannian curvature tensors;  
6.4. Theory of hypersurfaces in $\mathbb{R}^n$.


7. Tensor Analysis (3-4 weeks)

7.1. Some differential operators and the idea of taking average on manifolds;
7.2. Submanifolds, mean curvature vectors, and the Laplacian operator;
7.3. Exterior differentials and the divergence theorem;


8. Differential Structures of Riemannian Geometry (3-4 weeks)

8.1. Structure equations;
8.2. Covariant differentiation of tensors;
8.3. Commutation formulas and moving frames;  


9. The Method of Moving Frames and Global Differential Geometry (3-4 weeks)

9.1. Gauss-Bonnet-Chern Theorem;
9.2. Bochner's technique;  
9.3. Eigenvalues of Laplacian and Obata Theorem

 

 
Instructional Approach
Methods Notes
Formal lecture  
Group discussion  
Grading assessment
Methods Percentage Notes
Final exam 70 %  
Class discussion involvement 30 %  
Required and Recommended Texts/Readings with References

1. S. S. Chern, W. H. Chen, K. S. Lam, Lectures on differential geometry. Series on University Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 1999.

2. Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry. Third edition. Universitext. Springer-Verlag, Berlin, 2004. xvi+322 pp. ISBN: 3-540-20493-8

3. D. A. Lee, Geometric Relativity, Amer. Math. Soc., 2019.

4. J. M. Lee, Introduction to Riemannian manifolds. Second edition. Graduate Texts in Mathematics, 176. Springer, Cham, 2018.

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