Honors Differential Geometry, Math 5540H (class number 35535),
Spring Semester 2024

Instructor: Sergei Chmutov

Classes: Monday 2:15--4:05 pm. Wednesday, Friday 2:20--3:40 pm. The classes will be in-person in Journalism Bldg 304. All the classes will be recorded through ZOOM and the the link will be posted on Carmen.

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 which might be used:

  • 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.

    Homework will be assigned weekly through Carmen due Wednesday. You are supposed to upload your solutions to Carmen. Penalty for late submission is 10 points per day.

    Exams: There will be two midterms and one final exams. You may NOT use your calculator, text, notes, or solutions to assignments. Except for extraordinary reasons, you must take the examinations at the scheduled times.

    TENTATIVE EXAMINATION SCHEDULE:
    MIDTERM 1 Friday, February 2. MIDTERM 2 Friday, March 8. FINAL EXAM Tuesday, April 30, 4:00 - 5:45 PM

    Grading: Homework assignments average (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. Frenet equations. Global properties of plane curves. Curves in R3 and Rn.
    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. Multilinear algebra.
          Tensors. Vector fields and bundles. Differential forms.
    4. Intrinsic geometry of surfaces.
          Riemannian metrics. Gauss-Bonnet theorem.
          Parallel transport. Geodesics. Christoffel symbols.
    5. Introduction to the calculus of variations.
          Variations of arc length. Euler-Lagrange equation.
        & Jacobi fields and conjugate points.

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