#

MAT327 Introduction to Differential Geometry

Fall 2007

## Course Information

** Time: M W 1:30 p.m. - 2:50 p.m.**

** Classroom: Fine 1001**

** Instructor: ** Anna Wienhard, email

**Office:** Fine 1007

** Office Hours:** Monday 3:00 p.m. - 4:00 p.m.

** Grader:** Ye Li, Fine 311, email

**Textbook**: *Differential Geometry of Curves and Surfaces*, by Manfredo P. do Carmo.

** Topics:**: Curves and Surfaces in R^{3}

**Syllabus**

**Grading:**

Take Home Midterm Exam 20 %

Take Home Final Exam 30 %

Homework Assigments and Project 50 %

## Assignments

The homework assigments will be posted a week before the due date. Unless noted otherwise all exercises are from the Textbook.

**Homework, due September 24:**
Read Handouts on "Existence and Uniqueness of Solutions to ODE's" and "Inverse and Implicit Function Theorems".

Parametrized curves and arc length: Problem 2 on page 7, Problem 6 on page 8/9, Problem 8 on page 10/11.

Curvature and Torsion: Problem 1 on page 22, Problem 6 on page 23, Problem 12 on page 25.

Not required: Problem 17 on page 26 (help is in the back of the book).

**Homework, due October 1:**
Regular Surfaces: Problem 1 on page 65, Problem 3 on page 65, Problem 7 on page 66, Problem 12 on page 66, Problem 16 on page 67, Problem 18 on page 68

Not required: Read Example 6 on page 65 and find three coordinate neighborhoods which cover the torus.

**Homework, due October 8:**
Assigment

**Homework, due October 15:**
Assigment

**Homework, due October 22:**
Assigment

**Homework, due November 5:** No Assigment, due to Midterm and Fall Break. You should look at the examples in Chapter 3-3 of doCarmo and review what we did so far in class.

**Homework, due November 12:**
Problem 1 on page 151, Problem 22 on page 173, Problem 3 on page 228, Problem 4 on page 228, Problem 9 on page 229, Problem 10 on page 229, Problem 16 on page 230.

Start working on your project.

**Homework, due November 19:**
Problem 10 on page 187, Problem 3 on page 237, Problem 7 on page 237, Problem 8 on page 237.

Problem: Let ** x** (u,v) = (f(v) cos(u), f(v) sin(u), g(v)) be a local parametrization of a surface of revolution such that the curve (f(v), g(v)) is parametrized by arc length. Compute the Christoffel symbols of the surface of revolution in this parametrization.

Work on your project.

**Homework, due November 26:**
Problem 1 on page 237, Problem 2 on page 260, Problem 5 on page 260, Problem 7 on page 261, Problem 13 on page 261, Problem 15 on page 262.

Continue to work on your project.

** Homework, due December 3:**
Problem 9 on page 261, Problem 11 on page 261, Problem 14 on page 261, Problem 18 on page 262, Problem 20 on page 263.

Work on our project! (You have to hand it in on December 5!)

**Homework, due December 10:**
Problem 1 on page 282, Problem 2 on page 282, Problem 3 on page 282, Problem 5 on page 282/283, Problem 23 on page 264.

## Handouts

Metric Spaces (from MIT Open Courseware)

The Contraction Mapping Theorem (from MIT Open Courseware)

The Existence and Uniqueness of Solutions to ODE's (from MIT Open Courseware)

Inverse and Implicit Function Theorems (from MIT Open Courseware)

## Pictures of Curves and Surfaces

Virtual Math Museum

Visual Dictionary of Special Plane Curves

Gallery of Famous Surfaces

Wolfram MathWorld Curves

Wolfram MathWorld Surfaces

Differential Geometry Images

Bianchi Website (with Mathematica programs for download)