Date    Chapter.Section/Topics   or   Page/Problems

Week 1
M   Jan 8  Lecture 1
           (see "Additional Materials" on main course page for handouts)
W    10    Lecture 2
           Example C
Week 2
M    15    holiday
W    17    Minimizing length fun(ctionals)
Week 3
M    22    Solution of Exercise 3.1
           Topological spaces
           Poincare manifolds
           Lie groups warm up
W    24    Parametric variations of Euclidean arclength
Week 4
M    29    Bondurant on B
           Other thoughts on B
           More solutions (Exercise 7.1 on locally Euclidean topological manifolds)
W    31    
Week 5     A question and an answer
M  Feb 5   
W     7    Poincare conjecture (updated)
           A question and an answer (updated/corrected)
Week 6
M    12    Manifold B (updated with linear structure)
           Spaces (linear and vector spaces)
           More thoughts on B (Lance's question)
W    14    Some material related to the suggested calcuation 
           given on this date are contained in the postings for 
           February 18 below.
Week 7
M    19    Cayley geometry exercise
           ODE for geodesics and geodesics in B
W    21    Linear structure/atlas notes (not much but it's what I've got typed at the moment)
Week 8
M    26    
W    28    Toplogical dimension and the Hausdorff condition (John Stavroulakis)    
Week 9
M  Mar 4   ODE for geodesics and geodesics in B
W     6    Python script for visualization of geodesics in B  (Zihao Chen)
Week 10 
M    11    Metric tensor and big question:  What is the intrinsic derivative?  
           calculus differentiation review
           metric tensor notes
           Coming soon:  Notes on your questions
           (a) John S. Singular Lagrangians?
           (b) Lu L. Second derivatives?
           (c) Sergey B. Are (the) linear Liebnizian functionals realizable as a linear dual space?   
W    13    Intrinsic derivative of f:M --> R
Week 11
M    18    Spring Break
Tu   19    ODE for geodesics and geodesics in B
W    20    Spring Break
Week 12 
M    25    The intrinsic gradient
W    27    Calculating some intrinsic derivatives of functions
           derivative of f (updated)
           vector fields (draft)
Week 13
M  Apr 1   Intrinsic Gradient (Ruijia)
           Janet's isometric embedding theorem
           some solutions
W     3    resources for isometric embedding
           My notes on Janet's theorem
M     8    Janet's theorem and flexibility
W    10    Intrinsic gradient of coordinate functions on S^2
           vector fields (update)
M    15    Intrinsic derivative of a vector field
           Example calculations in Euclidean space
W    17    Example calculations continued
           Outline for the semester
            a. Riemannian metric tensor (tangent spaces etc.)
            b. intrinsic derivatives of functions
            c. intrinsic derivatives of vector fields
            d. curvature (at your liesure)
            e. conjugate points/structure of geodesics/theorems
           student questions
           bibliography
           Existence of Willmore minimizers
           Bryant Duality
M    22    Comments about curvature
W    24    Comments about curvature
F    26    Final Problems