Week #1

Sept 26-30

 Curves:
  • Parametrized regular curves; Arc length;
  • Cross product, curvature and Frenet formulas.
       Problem 7, p. 23
  • Isoperimetric inequality.
    		• September 26th: classes begin
    Week #2

    Oct 3-7

    		Regular surfaces.
  • representing surfaces as: images of parametrization, graph of function or inverse image of a point.
  • Local coordinates on the surface. Surfaces of revolution.
  • Differentiable functions on surfaces; tangent planes. Oreinted surface. Normal vector. Tangent surface of a curve (p. 78)
    • Week #3

    Oct 10-14

  • 2.5: The first fundamental form (E, F, G coefficients of the metric). Area of the surface.
  • 3.2: Gauss map. Second fundamental form. Normal, Gaussian and mean curvature.
  • Line of curvature (p.145). Elliptic, hyperbolic and parabolic points on a surface.

    • Oct 14th: Last day to add or drop a class.
    Week #4

    Oct 17-21

  • 3.3: Local coefficients of the second fundamental form. Asymptotic direction.
  • Curvature of a surface given as graph of function. Example 5 p. 162
  • Geometrical meaning of Gauss map (p. 166- 167)
  • 3.4: Vector fields. Derivative along a vector field. (Pb. 7)
    • Week #5

    Oct 24-28
  • 3.5 A: Ruled surfaces.
  • 3.5 B: Minimal surfaces.
  • 4.2: Local v.s. global isometries.
    		• Oct 26th: Midterm.
    Week #6

    Oct 31-Nov4

  • 4.3: Christophel symbols. Gauss Theorem Egregium.Mainardi-Codazzi equations (p.235)
  • 4.4: Parallel transport. Covariant derivative.
  • Geodesics in terms of covariant derivative.
    • Week #7

    Nov 7-11

  • Algebraic expression for covariant derivative (p. 252)
  • Liouville's Theorem. Uniquness of geodesic in a given direction.
  • 4.5: Theorem of Turning Tangents.
    		• Nov 7th: Term withdrawal deadline.
    Week #8

    Nov 14-18
  • Gauss-Bonnet Theorem. Euler characteristics.
  • Global Gauss-Bonnet.
  • The Exponential Map.

    • Nov 18th: Change of grading basis deadline.
    Week #9

    Nov 28-Dec 2
  • 5.3: Hopf-Rinow Theorem.
  • 5.4: Geodesics as length minimizing curves.
  • Bonnet's Theorem.
    • Week #10

    Dec 5-9

  • 5.7: Fary-Milnor Theorem.
    		•  End-Quarter Period.
    Final Exam December 16th 8:30-11:30 AM