## Spring 2000, Spring 2003, Spring 2008, Spring 2010

Math346: Differential Geometry II. A fourth year course.

Text Book : Differential Geometry of Curves and Surfaces, by
Manfredo P. do Carmo. Prentice-Hall, New Jersey , 1976.

Other books : Barret O'Neill, Elementary Differential Geometry, Academic Press (second edition)
Differential Geometry ,
Curves
Surfaces1 , Surfaces2 , symmetry
Content : In this period the remaining last two chapters of " Manfredo P. do Carmo " will be covered. See the "Course Syllabus" below
Course Schedule
• Monday 8.40-10.30 , SAZ-19
• Wednesday 10.40-12.30 , SAZ-19
Exams
• (30%) First Midterm Exam pdf file (2003) , (2008) ,, (2010) , : March 17. (10.40-12.30) SAZ19 .
• (30%) Second Midterm Exam pdf file (2003) , (2008) , (2010) : April 28. (10.40-12.30) SAZ19 .
• (30%) Second Midterm Exam pdf file (2003) , (2008) , (2010) : April 28. (10.40-12.30) SAZ19 .
• Makeup exam (2010) : May 31, 2010 (10.00-12.00) SAZ18 .

• homework and exam grades (2003)

You are responsable from the sections of Chapter 4 which were covered by Prof. Degtyarev.
I will review the sections 5.1,5.2 and 5.3 which were also covered by Prof. Degtyarev.

ASSIGNED EXERCISES:

1. Geodesics and Parallel Transport (Section 4.4 of do Carmo).

2. (The Holonomy Group) 1. Exercise 22 in Section 4.4 and 2.Exercise 14 in Section 4.6 (do Carmo)

3. Exponential Mapping (Section 4.6 of do Carmo).

4. Gauss-Weingarten-Codazzi Equations

5. Isometry, Infinitesimal Isomtery and Killing Vectors

6. Calculus of Variations

Subjects covered

1. February 1
• The Rigidity of the Sphere (Chapter 5)
• Complete surfaces and Hopf-Rinow Theorem
2. February 8
• First and Second Variation of Arc Length
• Bonnet`s Theorem
3. February 15
• Jacobi Fields and Conjugate points
4. February 22
• Jacobi fields
5. March 1
• Jacobi Fields
6. March 8
• Covering spaces
7. March 15
• First Midterm Exam
8. March 22
• Covering spaces and Hadamard Theorems
9. March 29
• Synge's Lemma
• Some global theorems for closed curves
10. April 12
• Jordan closed curve theorem
• Killing Vector fields
11. April 19
• Fenchel and Fary-Milnor theorems
• Calculus of variations in Differential Geometry
12. April 26
• Fary-Milnor Theorem
• Calculus of variations in Differential Geometry
Second Midterm Exam
13. May 3
• Calculus on Euclidean Space (Chapter 1 of Peter O'Neil)
14. May 10
• Frame Fields (Sections 2.5-2.8, 6.1-6.2 of Peter O'Neil)

• Course Syllabus