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## MATH 346 - DIFFERENTIAL GEOMETRY II

## 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 __

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