MA489 Differential Geometry of Curves and Surfaces (Reading Course)

## Links

- 3dxplormath has a great library of plane curves, space curves, etc. I believe you can also see parallel curves, involutes, & evolutes of curves, and probably more
- TED Talk including a segment on crochet and non-Euclidean geometry (at about 9min)

# Text and Notes

Pressley, *Elementary Differential Geometry*

## Suggested Problems

**Chapter 1**(Parametrizations)- 1, 3-4, 5 (mathematica), 6, 9; 11; 14, 15; 17, 19
**Chapter 2**(Curvature)- 1i, 1iv; 3, 6, 7-11, 13; 14ii, 16, 17, 20, 21, 22
**Chapter 3**(Global Properties)- 1, 3; 5; 6, 7, 8
**Chapter 4**(Surfaces)- 1, 4, 5; 6, 8, 9, 10; 13, 15, 16; 18, 21; 25, 26, 27; 30
**Chapter 5**(The First Fundamental Form)- 1, 2, 3; 5, 8; 9, 11, 13, 14; 15, 16; 18, 19, 20
**Chapter 6**(Curvature)- 1, 2; 5, 6, 7, 9, 12; 15, 18, 22; 23, 24
**Chapter 7**(The Gauss Map)- 1, 2, 3, 5, 8, 9; 11; 14; 15, 17; 19
**Chapter 8**(Geodesics)- 1, 2, 5; 6, 7, 8, 9, 10; 14, 15, 18; 19; 21

## Notes

**13-Jan**: snapshot of hyperbolic/Euclidean/spherical geometries; sketches of parametrizations and curves in space PDF notes**15-Jan**: parametrization by arc length; preview of frames for space curves PDF notes**19-Jan**: frames and rigid motions in the plane; affine transformations PDF notes**21-Jan**: notes on frames for plane and space curves; questions on constant curvature and torsion slides and PDF notes**27-Jan**: here is Andrew Plucker's thesis on the lead factor in pursuit-evasion games**27-Jan**: notes on constant curvature/torsion, and on pursuit strategies PDF notes**2-Feb**: notes on the calculus of variations, plus proofs of the isoperimetric inequality and the 4-vertex theorem PDF notes 1 and PDF notes 2

**10-Feb**: notes on the definition of surfaces and smooth surfaces PDF notes**12-Feb**: notes on smooth surfaces and tangent spaces PDF notes**17-Feb**: notes on general classes of surfaces, quadric surfaces, and quadratic forms PDF notes**23-Feb**: notes on the first fundamental form and differential forms PDF notes**25-Feb**: notes on conformal maps and surface area PDF notes**27-Feb**: equiareal maps and surface curvature PDF notes

**3-Mar**: notes on the second fundamental form and the curvature of curves on a surface PDF notes**9-Mar**: notes on principal curvature and the classification of surface points PDF notes**24-Mar**: review of fundamental forms and principal curvature; Gaussian and mean curvatures PDF notes**26-Mar**: the pseudosphere; flat surfaces PDF notes and mathematica for pseudosphere/tractrix**30-Mar**: parallel surfaces; positive curvature on compact surfaces PDF notes**1-Apr**: the Gauss map PDF notes**3-Apr**: geodesics and their properties PDF notes**7-Apr**: geodesics on surfaces of revolution; angular momentum PDF notes**9-Apr**: length-minimizing properties of geodesics and geodesic coordinates PDF notes

**13-Apr**: definition of manifold PDF notes**15-Apr**: compactness and the Brouwer fixed point theorem PDF notes**23-Apr**: general definition of manifolds PDF notes**27-Apr**: manifolds and Riemannian metrics PDF notes**30-Apr**: hyperbolic geometry PDF notes

# Homework

**Assignment 1, due 2-Feb**- A) Write up problems 3, 7, 9, 16, 21 in Chapter 2.

B) In Mathematica, use the`Manipulate`command to plot the evolute of $y=x^n$ for various values of*n*. **Assignment 2, due 19-Feb**- A) Write up problems 1, 4, 6, 10 in Chapter 4

B) In Mathematica, plot the hyperboloid of one sheet together with several choices of straight lines, as given in problem 4.4. **Assignment 3, due 2-Mar**- A) Write up problems 13, 15, 21 in Chapter 4

B) Write up problems 1i, 1ii, 2, 8 11, 14, 16 in Chapter 5

C) In Mathematica, use the`Manipulate`command to plot the isometric deformation of the catenoid into the helicoid (see problem 5.8 on page 106). **Assignment 4, due 7-Apr**- A) Write up problems 1, 5, 9, 12, 15, 22, 24 in Chapter 6

B) Write up problems 1, 3, 8, 9, 14, 15 in Chapter 7 **Assignment 5, due 15-Apr**- A) Write up problems 1, 2, 7, 9, 14, 15, 19 in Chapter 8

# Special Topics

- Spacetime and general relativity
- Riemannian geometry
- Spherical/hyperbolic geometry
- Calculus of variations; paths of least action
- Minimal surfaces
- Invisibility and cloaking
- Pursuit manifolds

page revision: 81, last edited: 04 May 2009 14:55

## Comments and Questions