# Math 4250 Archived Notes

This page contains archived course materials from a version of the MATH 4250 course taught prior to 2010 from Do Carmo’s Differential Geometry of Curves and Surfaces book. These are not current course notes or homework assignments.

Notes from Do Carmo’s book:

1. Introduction and Overview .
2. The Frenet Frame .
## Homework 1.
3. Curves of constant curvature and torsion .
4. The Bishop Frame .
5. Link, Twist, and Writhe . Correction: The integrals for Link and Writhe should be multiplied by 1/(4 pi) in these notes. Thanks to Matt Mastin for pointing this out!
6. The Four-Vertex Theorem .
## Homework 2.
7. The Fabricius-Bjerre Theorem .
8. An Introduction to Integral Geometry .
9. Integral Geometry II .
10. Stuff turning inside out .
## Homework 3.
11. Introduction to Regular Surfaces .
12. Regular Surfaces as Level Sets of Smooth Functions .
13. Tangent planes and differentials. Review of quadratic forms. .
14. The first fundamental form. How to measure lengths, angles, and areas in the uv plane.
## Homework 4.
15. The Gauss map and the second fundamental form. Defining the form.
16. The geometric meaning of the second fundamental form. The definition of Gauss and Mean curvature.
## Homework 5.
17. The meaning of the second fundamental form, part II. Umbilic points, Asymptotic and Conjugate directions, the Dupin indicatrix. &lt;br /&gt; Sorry about the scan quality– these came out really light.
18. Computing with the second fundamental form in local coordinates. e, f, g, formulas for Gauss and Mean curvature.
19. Extracting geometric information from the Second Fundamental Form. Differential equations for asymptotic curves and lines of curvature. Surfaces of revolution. Graphs.
## Homework 6.
20. Isometries, proof that Helicoid and Catenoid are isometric. There is no scan of these notes. Read 4.1 in DoCarmo.
21. The Christoffel symbols, proof of the Theorema Egregium, Mainardi-Codazzi equations and Gauss Formula, compatibility equations and theorem of Bonnet.
22. Theory of Geodesics, Geodesics on Surfaces of Revolution. Note: This is really different than the corresponding chapter in DoCarmo, so you won’t be able to match it up as easily as you did the previous lecture notes.
## Some examples of geodesics near a black hole