Honors Differential Geometry, Math 5540H (class number 31019),
Spring Semester 2014

Instructor: Sergei Chmutov

Classes: Monday 2:15--4:05 pm. Wednesday, Friday 2:20--3:40 pm. at Caldwell Lab 0171.

Carmen: https://carmen.osu.edu

TEXTBOOKS:

  • Manfredo do Carmo: Differential Geometry of Curves and Surfaces.
    Prentice-Hall, 1976, ISBN:9780132125895.
  • Michael Spivak: A Comprehensive Introduction to Differential Geometry, Vol.1,2.
    Publish or perish, 1999. ISBN Vol.1:0914098705, ISBN Vol.2:0914098713.

    Additional books:

  • Barrett O'Neill: Elementary Differential Geometry.
    Revised Second Edition, Academic Press (Elsevier), March 2006.
  • Wolfgang Kühnel: Differential Geometry: Curves - Surfaces - Manifolds.
    Second Edition, AMS Student Mathematical Library Series Vol. 16, 2006.
  • Dmitry Fuchs and Serge Tabachnikov: Mathematical Omnibus: Thirty Lectures on Classic Mathematics. AMS, 2007.
  • Shlomo Sternberg: Curvature in Mathematics and Physics. Dover, 2012.
  • Charles Misner, Kip Thorne, and John Wheeler: Gravitation. W. H. Freeman and Company, 1973.

    Grading: There will be weekly homework assignments (25%), two midterms (25% each), and a final exam (25%).
    GRADING SCALE:
    A A- B+ B B- C+ C C- D+ D
    90 87 83 80 77 73 70 67 63 60

    TOPICS:
    1. Curvature of curves.

          Plane curves. Curves in 3D. Frenet equations.
    2. Surfaces.
          Curves on surfaces, Euler theorem, Meusnier theorem.
          Curvature of surfaces. Gauss Map. The first and second fundamental forms.
          Gaussian curvature. Egregium theorem.
    3. Intrinsic geometry of surfaces.
          Riemannian metrics. Gauss-Bonnet theorem.
          Parallel transport. Geodesics. Christoffel symbols.
    4. Introduction to the calculus of variations.
          Variations of arc length. Euler-Lagrange equation.
        & Jacobi fields and conjugate points.

    The additional topics which we may disscuss during the course.
          Vector fields and bundles. Tensors.
          Differential forms. General relativity theory.                


    HANDOUTS

    Handout #1: Curvature of plane curves.
    Handout #2: Global properties of plane curves.
    Handout #3: Curves in R n.
    Handout #4: Multilinear algebra.
    Handout #5: Tensors.
    Handout #6: Tensor fields.
    Handout #7: Calculus of differential forms.
    Handout #8: Vector fields and differential operators.
    Handout #9: Christoffel symbols and Riemannian metric.


    HOMEWORK

    Assignment #1 (due Monday, January 13, 2014)
    Assignment #2 (due Wednesday, January 22, 2014)
    Assignment #3 (due Monday, January 27, 2014)
    Assignment #4 (due Monday, February 3, 2014)
    Assignment #5 (due Monday, February 17, 2014)
    Assignment #6 (due Monday, February 24, 2014)
    Assignment #7 (due Monday, March 3, 2014)
    Assignment #8 (due Monday, March 17, 2014)
    Assignment #9 (due Monday, March 24, 2014)
    Assignment #10 (due Monday, March 31, 2014)
    Assignment #11 (due Monday, April 14, 2014)
    Assignment #12 (due Monday, April 21, 2014)