MATH 3041

DIFFERENTIAL GEOMETRY

4+0

Prerequisites: None

Text book: Differential Geometry of Curves and Surfaces, M. P. Do Carmo. Elementary

Differential Geometry, A. Pressley.

Reference Books: Theory and Problems of Differential Geometry, M. M. Lipschutz,

Elementary Differential Geometry, B. O’Neill.

Course Objectives: The course is aim to introduce the differential geometry of curves and

surfaces both in their local and global aspects. The prerequisites for understanding this course

are linear algebra and calculus. Especially, students are expected to be familiar with calculus

of several variables including the statement of the implicit function theorem and the inverse

function theorem.

Course Outline:

1. Curves (5 weeks)

Parametrized curves, Regular curves, Arc length, Reparametrization, the vector product in

R^3, The local theory of curves parametrized by arc length, The local canonical form,

Arbitrary speed curves, Global properties of plane curves.

2. Continuity and Differentiability (1 week)

3. Regular Surfaces (3 weeks)

Inverse images of Regular Values, change of Parameters, The Tangent Plane.

4. The First Fundamental Form, Area (1 week)

5. Orientation of Surfaces (2 weeks)

A characterization of Compact Orientable Surfaces

GRADING:

There may be some presentations which will be given by students. The presentations will

affect the grading 15%. For the mid-term exam, there will be a written assignment. It will

count for 35% of the grading. The final exam will be 50% of the grading.