Differential Geometry

TuTh 11:00am-12:15pm in MSB303
Syllabus

Office: MSB M312    Phone: (860) 486-3850     e-mail: kaufmann@math.uconn.edu
Office hours:  TBA and by appointment

Course description

The basic idea and of differential geometry is to say something about the geometry of an object by moving a little bit on this object - for instance moving along a curve or on a surface. Turning this approach into a questions it reads: what kind of information can I get about my curve or my surface if I move a little bit on them? It turns out that there is indeed a lot one can learn. For a curve, one gets tangent directions, curvature and other geometric information in this way. For a surface, there are 2-d generalizations of these concepts.
One striking fact is that knowing this information everywhere allows you for instance to discover that the earth is not flat. Furthermore it allows to explain why there cannot be any maps of the earth which give the right distances and angles at the same time. These types of considerations are also the basis for the theory of general relativity.

In this course, we will treat curves and surfaces from the above perspectives which lead us to the results discussed above. We will provide a classical treatment, but the results and concepts have applications in discretized versions for computer imaging and methods of finite elements.

I will try to encorporate computer animations if possible. If all goes well we should also have the time to discuss
special and a little bit of general relativity at the end of the course.

News: Sample final is up: Look at it, we will discuss it on Th.

Sample midterm
Sample final

Homework

 Chapter/Section Numbers Due HW0 (optional) 1-4 Re-read e.g. Steward Multivariable calculus 5e Chapter 13 and do some of the exercises 1-2 1b 1,4,5 HW1 1-3 1-5 4,6 1, 10 a,b) 12* Sep 19 HW2 2-2 1,2,4,10,16 Oct 17 HW3 2-3 3, 4, 6, 11 Nov 7 HW4 2-4 3, 9, 10, 11, 13 (optional) Nov 14 HW5 2-5 2-6 3, 9, 10, 11 2, 5 (harder) Nov 30 Assignment 1 Prepare a ca 5-10min presentation on "Your Favorite Plane Curve"** Sep 28 Assignment 2 Write an overview of the maping techniques/functions used for the earth. *** Oct 26

*means optional
**The basic idea is that you write down the equations for a curve, give its geometric properties, relate them to the equation and maybe give history and/or applications.
Check the famous curve index for a starting point. You can also pick an example from the book or come up with something on your own.
Other references can be found in the library or at Mathworld and Wikipedia.
If you would like, you can work in groups of up to 2 people (you do not have to).
***You can for instance check the first pages of any good atlas for a start.

Supplemental material: Linear Algebra, Differentiable Maps, Jacobians and the Chain Rule
Background Information on Topology (Advanced material which gives an introduction to topology)