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. **

- Crofton’s Formula and Buffon’s Needle.
- Crofton’s Formula and the Indicatrices.
- The Four Vertex Theorem.
- The Bishop Frame.

Notes from Do Carmo’s book:

- Introduction and Overview .
- The Frenet Frame .

## Homework 1. - Curves of constant curvature and torsion .
- The Bishop Frame .
- 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! - The Four-Vertex Theorem .

## Homework 2. - The Fabricius-Bjerre Theorem .
- An Introduction to Integral Geometry .
- Integral Geometry II .
- Stuff turning inside out .

## Homework 3. - Introduction to Regular Surfaces .
- Regular Surfaces as Level Sets of Smooth Functions .
- Tangent planes and differentials. Review of quadratic forms. .
- The first fundamental form. How to measure lengths, angles, and areas in the uv plane.

## Homework 4. - The Gauss map and the second fundamental form. Defining the form.
- The geometric meaning of the second fundamental form. The definition of Gauss and Mean curvature.

## Homework 5. - The meaning of the second fundamental form, part II. Umbilic points, Asymptotic and Conjugate directions, the Dupin indicatrix. <br /> Sorry about the scan quality– these came out really light.
- Computing with the second fundamental form in local coordinates. e, f, g, formulas for Gauss and Mean curvature.
- Extracting geometric information from the Second Fundamental Form. Differential equations for asymptotic curves and lines of curvature. Surfaces of revolution. Graphs.

## Homework 6. - Isometries, proof that Helicoid and Catenoid are isometric.
*There is no scan of these notes. Read 4.1 in DoCarmo.* - The Christoffel symbols, proof of the Theorema Egregium, Mainardi-Codazzi equations and Gauss Formula, compatibility equations and theorem of Bonnet.
- 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

## (pdf version of the relativity examples)

## Homework 7. - (Signed) geodesic curvature and the local Gauss-Bonnet theorem. (These notes have everything there, but could make more sense. Read with a bit of caution. The reference here is McCleary, Geometry from the Differentiable Viewpoint, p. 173-177.)
- The global Gauss Bonnet theorem, Euler characteristic, applications of G-B, conclusion.