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

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