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 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

Comments and Questions