Differential Geometry of Curves and Surfraces
Math 436 Fall 2009

Official Course Number: MATH 436 (Section 0101) (3 credits)
Grade Method: REG/P-F/AUD.
Lectures: Tuesday-Thursday 9:30 - 10:45, B0421 Math Building
Professor: Bill Goldman (wmg@math.umd.edu)
Office: 3106 Math Building
Phone: 301-405-5124
Office hours: Tuesday/Thursday 1:00- 1:45 or by appointment.

Prerequisites: MATH241; and either MATH240 or MATH461; and two 400-level MATH courses (not including MATH400, 461 and 478).

Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, "Theorema Egregium"; the intrinsic geometry of surfaces.

Text: Elementary Differential Geometry, First Edition, by Andrew Pressley, Springer Undergraduate Mathematics Series, ISBN 1-85233152-6 (2001)

Tentative Syllabus

  1. Curves
    1. Plane curves, signed curvature
    2. Space curves
    3. Curvature and torsion, fundamental theorem
  2. Regular surfaces
    1. Implicit function theorem
    2. Tangent planes, linearization
  3. Extrinsic geometry of surfaces
  4. Intrinsic geometry of surfaces

Tentative Schedule

  • First problem set: 1.14, 2.2, 2.7-2.9 (plane curves), due Thursday 9/10
  • Second problem set: 2.14, 2.15, 1.9
  • Problem Session: Tuesday, September 22.
  • Third problem set: 4.3, 4.8, 4.17, 4.18, 4.19, 4.28
  • The first test was given Tuesday, October 13.
  • Fourth problem set: 5.1, 5.2, 5.3, 5.4, 5.5, 5.8, 5.11, 5.13, 5.17, 5.19
  • Write up a more conceptual proof of Theorem 5.2 (pp. 108-109) along the lines of the lecture on Thursday, November 5.
  • Read Chapter 6. Fifth problem set: 6.7, 6.9, 6.10, Problems 6.11-6.14 (on geodesic torsion) are recommended, but optional.
  • Read Chapter 7.
  • Read Chapter 8, 10, 11.
  • The final text will be given Thursday, December 11.

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Last modified: 2 December 2009