Introduction to Differential Geometry 2
(Fall, 2017) 화목 15:30-16:45 (24동 209호)
We will study and understand Gauss-Bonnet Theorem
and Morse theory for surfaces.
References
- M. do Carmo, Differential Forms and Applications, Springer, 1994.
- M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., 1976.
- O'Neil, Elementary Differential Geometry, Academic Press, 2nd ed., 1997.
- T. Banchoff, S. Lovett, Differential Geometry of Curves and Surfaces,
A K Peters/CRC Press, 2010.
- A. Fomenko, T. Kunii, Topological Modeling for Visualization, Springer, 1997.
- A. Pressley, Elementary Differential Geometry, 2nd ed., Springer, 2010.
(앤드류 프레슬리 지음, 전재복, 채영도, 김병학 옮김, 미분기하학 입문, 경문사, 2009)
- J. Oprea, Differential Geometry and its applications, Prentice Hall, 1997.
- J. McCleary, Geometry from a Differentiable Viewpoint, Cambridge, 1994.
- http://www.math.snu.ac.kr/~hongjong/IDG-Ref.html
Syllabus
Week 1:
What is a Euclidean space?
What are the motions?
Week 2:
What is a surface?
Week 3:
What is a manifold? What is a Riemannian manifold?
Week 4:
Euler characteristic and Classification of Surfaces
Week 5:
First and Second Fundamental Forms
Week 6:
Fundamental theorem for the theory of surfaces
Week 7 (Oct. 24, 화) :
Midterm
Week 8:
Principal Curvatures,
Gauss Map, Gaussian Curvature, Theorema Egregium
Week 9:
Geodesics
Week 10:
Surfaces of Constant Curvature
Week 11:
Trigonometry
Week 12:
Vector fields and indices
Week 13:
Gauss-Bonnet Theorem
Week 14:
Morse Theory
Week 15 (Dec. 12, 화): Final Exam
Evaluation
Midterm 30%,
Final 40%,
Homeworks 20%,
Etc. 10 %