Math 206: Differential Geometry (Spring 2010)

This course will present an introduction to differential geometry of curves and surfaces in 3-space. Topics to be covered include first and second fundamental forms, geodesics, Gauss-Bonnet theorem, and minimal surfaces. Applications to problems in math, physics, biology, and other areas according to student interest.

Prerequisites

Math 103 or Math 105 Multivariable Calculus, Math 104 Linear Algebra.

Schedule

Mon/Wed/Fri, 8:45-9:35am, Room 205 Physics Building

Instructor

Jian-Guo Liu, Professor, Dept of Physics and Dept of Math
Office: Room 285 Physics Building, 660-2546, jliu "at" math.duke.edu

Office hours

Monday and Friday 9:35am-noon, room 285 Physics, or by email request for an appointment for other times.

Textbook

Elementary Differential Geometry, by B. O'Neill, Academic Press, Revised second edition, 2006.

Grading

The final exam (May 7, Friday, 9am-noon) will constitute 30% of the grade.
Two in-class midterm exams (The 1st midterm will be held on Feb 24 Wednesday. The date of the 2nd midterm will be determinted) togeter will count 40% of the final grade (20% each)
Weekly homework assigments will constitute 30% of the final grade. The lowst homwwork grade over the semester will be dropped before determining the final grade

Problem sets

Homework Assigment 1, due Jan 20 Wed

§1.1: 1, 2, 3, 4

§1.2: 1, 3

§1.3: 3, 4, 5

Homework Assigment 2, due Jan 27 Wed

§1.4: 2, 4, 8

§1.5: 1, 2, 4, 7, 9, 10

Homework Assigment 3, due Feb 3 Wed

§1.6: 1, 3, 4, 5, 7, 8, 9

§1.7: 5, 6, 7, 9, 10

Homework Assigment 4, due Feb 10 Wed

§2.1: 4, 12

§2.2: 1, 2

§2.3: 2, 5, 8, 10

Homework Assigment 5, due Feb 17 Wed

§2.4: 2, 4, 6, 10, 12

§2.5: 2, 4, 5

§2.6: 2

Homework Assigment 6, due Feb 24 Wed

§2.7: 1, 4, 5, 8

§4.1: 4, 6, 10, 12

Midterm exam 1, Feb 24 Wednesday, Chapter 1 and Chapter 2

Homework Assigment 7, due March 3 Wed

§4.2: 2, 3, 6a, 9

Homework Assigment 8, due March 17 Wed

§4.3: 3, 4, 8, 12

§4.4: 1, 3, 4, 7

Homework Assigment 9, due March 24 Wed

§4.5: 3, 5, 8, 9, 10

§4.6: 2, 7, 8, 10

Homework Assigment 10, due March 31 Wed

§5.1: 1, 2, 3, 4, 7

§5.2: 1, 4

Homework Assigment 11, due April 7 Wed

§5.3: 2, 3, 6a

§5.4: 2, 3, 6, 12

Homework Assigment 12, due April 14 Wed

§5.6: 1, 2, 7

Midterm exam 2, April 14, Wednesday, Chapter 4 and Chapter 5

Homework Assigment 13, due April 21 Wed

§6.1: 1, 2

§6.2: 2

Homework Assigment 14, due April 28 Wed

§6.3: 1, 3

§6.4: 1, 3, 9, 14

§6.5: 1, 4

Final Exam, May 7, Friday, 9am -- noon

Reference books

Elementary Differential Geometry by Andrew Pressley, Springer Verlag, 2001
Differential Geometry and its Applications by John Oprea , Mathematical Association of America Textbooks, 2007
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo , Prentice Hall, 1976
Real Mathematical Analysis by C. C. Pugh , Springer, 2003

Syllabus

Chapter 1, Calculus on Euclidean Space

§1.1 Euclidean Space

§1.2 Tangent Vectors

§1.3 Directional Derivatives

§1.4 Curves in &real³

§1.5 1-forms

§1.6 Differential Forms

§1.7 Mappings

Chapter 2, Frame Fields

§2.1 Dot Product

§2.2 Curves

§2.3 The Frenet Formulas

§2.4 Arbitrary Speed Curves

§2.5 Covariant Derivatives

§2.6 Frame Fields

§2.7 Connection Forms

Midterm exam 1, Feb 24 Wednesday

Chapter 4, Calculus on a Surface

§4.1 Surfaces in &real³

§4.2 Patch Computations

§4.3 Differentiable Functions and Tangent Vectors

§4.4 Differential Forms on a Surface

§4.5 Mappings of Surfaces

§4.6 Integration of Forms

Chapter 5, Shape Operators

§5.1 The Shape Operator of M &sub &real³

§5.2 Normal Curvature

§5.3 Gaussian Curvature

§5.4 Computational Techniques

§5.6 Special Curves in a Surface

Midterm exam 2, April 14

Chapter 6, Geometry of Surfaces in &real³

§2.8 The Structural Equations

§6.1 The Fundamental Equations

§6.2 Form Computations

§6.3 Some Global Theorems

§6.4 Isometries and Local Isometries

§6.5 Intrinsic Geometry of Surfaces in &real³

Final Exam, May 7, Friday, 9am -- noon