Math 4250

Math 4250

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Welcome to the homepage for Differential Geometry (Math 4250/6250)! I will post all the homework assignments for the course on this page. Our text for the course is DoCarmo, which is a good text, but has a fair number of errors. See the electronic resources at the end of this page for a list of errors in the text-- if you are very confused, it is probably a good idea to check this list and make sure that the book is right!

Homework will be due on Thursdays, at more-or-less weekly intervals. Each assignment will contain 4 regular problems and 4 "challenge" problems. From these problems, you should pick five problems of your choice. Remember that undergraduates should average one challenge problem per assignment, while graduate students should average two challenge problems per assignment.

Contents

Lecture Notes and Homework Online

Here are links to my lecture notes for the course; these will posted as I can manage it, to provide a supplement for the text in case you're having trouble, or want another perspective on what we did in class. Some of these are LARGE scanned PDF files, so you probably don't want to download them at home.

  1. Introduction and Overview.
  2. The Frenet Frame.
    1. Homework Assignment 1. Due date is Tuesday, 1/23/7.
  3. Curves of constant curvature and torsion.
  4. The Bishop Frame.
  5. 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!
  6. The Four-Vertex Theorem.
    1. Homework Assignment 2. Due date is now Thursday, 2/8/7.
  7. An Introduction to Integral Geometry.
  8. Stuff turning inside out.
    1. Homework Assignment 3. Due date is Thursday, February 22.
    2. Take-home test 1. Now due on Friday, March 2, 2007 by 5pm (slip test under my door).
  9. Introduction to Regular Surfaces.
  10. Regular Surfaces as Level Sets of Smooth Functions.
  11. Tangent planes and differentials. Review of quadratic forms..
  12. The first fundamental form. How to measure lengths, angles, and areas in the uv plane.
    1. Homework Assignment 4. Due on Thursday 3/22.
  13. The Gauss map and the second fundamental form. Defining the form.
  14. The geometric meaning of the second fundamental form. The definition of Gauss and Mean curvature.
    1. Homework 5. Due on Tuesday, April 3.
  15. The meaning of the second fundamental form, part II. Umbilic points, Asymptotic and Conjugate directions, the Dupin indicatrix.
    Sorry about the scan quality-- these came out really light.
  16. Computing with the second fundamental form in local coordinates. e, f, g, formulas for Gauss and Mean curvature.
  17. Extracting geometric information from the Second Fundamental Form. Differential equations for asymptotic curves and lines of curvature. Surfaces of revolution. Graphs.
    1. Homework 6. Due on Tuesday, April 10.
  18. Isometries, proof that Helicoid and Catenoid are isometric. There is no scan of these notes. Read 4.1 in DoCarmo.
  19. The Christoffel symbols, proof of the Theorema Egregium, Mainardi-Codazzi equations and Gauss Formula, compatibility equations and theorem of Bonnet.
    1. Exam 2 Due on Thursday, April 19.
  20. 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.
    1. Homework 7. Due Monday, April 30.
  21. (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.)
  22. The global Gauss Bonnet theorem, Euler characteristic, applications of G-B, conclusion.
    1. The (delayed) final exam This will now be due on Friday (sorry for the delay in posting it!).

Please examine the course syllabus. If you think you can get by with this copy, save a tree! Don't print it out.

Practice Exams.

Two practice exams (for the first and second midterms) are available: 4250 Practice Exam 1 and 4250 Practice Exam 2

Wiki extra credit challenge

Find a topic from our class and add an entry to this wiki covering the entry in the style of Wikipedia, using your book as a primary source and other books, or papers as needed. Make your username your real name, and link to this page. Note that Google is very efficient at detecting material copied from other webpages or wikis without attribution, so please write your own entry from scratch.

Electronic Resources

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