Math 212, Differential Geometry
Not offered in 2005-2006 

Differential geometry of curves and surfaces has two aspects. One, which may be called classical differential geometry, started with the beginnings of calculus. Classical differential geometry is the study of local properties of curves and surfaces. By local properties we mean those properties which depend only on the behavior of the curve or the surface in a neighborhood of point. The other aspect is called global differential geometry: here we see how these local properties influence the behavior of the entire curve or surface.

The main idea is that of curvature.

• What is curvature? It can be intrinsic or extrinsic. What's the difference?
• What does it mean to say that one curve has larger curvature than another?
• What does it mean to say that a surface has positive or negative curvature?
• Can a surface have both positive and negative curvature?

We will answer these questions for surfaces in 3-space, and we will use the different types of curvature to classify surfaces of various types. In particular we will discuss minimal surfaces, a part of differential geometry that has seen explosive growth in the last decade or so, and that has some interesting applications.

The foundations of differential geometry are very concrete. We can explore the concepts of geodesics, and extrinsic and intrinsic curvature, etc. all in the familiar setting of R³. The weekly homework assignments will contain both computations and proofs. This course is a good follow-up to Math 206, especially for students who would like some more practice writing proofs before jumping to a 300-level course.

Prerequisite: 205 or permission of instructor
Distribution: Mathematical Modeling.

• Megan Kerr mkerr@wellesley.edu
• Department of Mathematics
• Date Created: November 3, 1995
• Last Modified: June 27, 2005
• Expires: June 30, 2006