Math 32a---Differential Geometry Spring 2002

Daniel Ruberman 310 Goldsmith Hall

email: ruberman@brandeis.edu Phone: 736-3074

Office Hours: Tuesday 10-12, Friday 11-12 and by appointment.

 

Differential Geometry is the study of how to measure the curvature of surfaces, curves, and higher-dimensional objects. Our main focus will be on surfaces in 3-dimensional space, using the tools of advanced calculus and linear algebra. These same tools will enable us to study higher dimensional analogues of surfaces (`hypersurfaces’). We will eventually learn that curvature, which seems to be a property of a surface as seen from the outside, is really an "intrinsic" notion, which depends only on the measurement of distance within the surface, not on how it sits in space. The ideas which will be developed in the course are the basis for General Relativity, as well as many other advanced topics in geometry.

Prerequisites: The more you know, the deeper we can go into the subject. The basic requirements are linear algebra (15/21a/22a) and multivariable calculus (20a/21b/22b); this course is a great opportunity to truly appreciate how powerful are the tools you acquired in those courses. One of our main theoretical tools will be the implicit function theorem, which will be reviewed if it is not familiar to everyone. We will also make use of some of the basic theory of ordinary differential equations (existence and uniqueness theorems). I will supply background on this as needed by the class.

Practical Stuff: There will be one midterm exam and a final. Homework will be collected regularly and will be graded. The best way to learn the material in this course is to do lots of exercises, to gain calculation skill and grasp the concepts, so I urge you to put in lots of time on the homework. The grade will be based on the exams and the homework.

Book and Course Outline: We will follow the text "Elementary Topics in Differential Geometry", by J.A. Thorpe. Some copies of the book are available in the bookstore; let me know if you have trouble getting a copy. I have placed a copy of the book on reserve in the Science Library. There are lots of interesting topics in the book; unfortunately we won’t manage to get to it all. I plan to cover at least Chapters 1-12, 14-15, and probably some additional topics, chosen from the later chapters of the book.

Additional References and Materials: A book which does some of the same subjects, but from a different point of view, is "Differential Geometry of Curves and Surfaces", by Mandfredo do Carmo. A copy of this book is on reserve in the Science Library. If you are familiar with Maple or Mathematica, you may find these programs useful in visualizing level sets of functions and implicitly given surfaces in 3-space. Later in the course, when we discuss parameterized surfaces, we can use Prof. Palais’ wonderful program 3D Filmstrip to visualize many complicated phenomena. The program, which runs on any Macintosh, has lots of neat stuff in it besides the parts which are relevant to this course. (If you have trouble downloading the program, try the mirror site in Germany .)