Professor: Eckhard Meinrenken, mein at math.toronto.edu
Office hours (SS4053): Thursday 9:30-10:30 or by appointment
(e-mail!)
Teaching assistant: Ho Minh Toan
Lectures:
MWF 3-4, MP118
Marking scheme: 6 Problem sets (60 %), final exam (40%).
Problem set #1:pdf format,
ps format. Due January 21 (in class)
Problem set #2:pdf
format,
ps format. Due February 4 (in class)
Problem set #3:pdf format,
ps format. Due February 25 (in class)
Problem set #4:pdf format,
ps format. Due Monday, March 8 (in class)
Problem set #5:pdf format,
ps format. Due Monday, March 22 (in class)
Problem set #6:pdf format,
ps format. Due Friday, April 2 (in class) EXTENDED TO MONDAY, APRIL 5
PROBLEM SET #6 IS NOW AVAILABLE FOR PICK-UP. (Or send me an e-mail, if
you just want to know your marks.)
Some remarks on the upcoming final exam: Most questions will
be computational: E.g. calculating torsion of a space curve, principal
curvatures of a surface, finding geodesics, etc. There may also be
theoretical questions where you are asked to properly state
definitions, theorems or short formulas - make sure you remember the
names. (E.g.: "State Frenet's formulas".) You won't be asked to
reproduce proofs from the lecture, or to state lengthy and complicated
formulas.
Textbook:
Andrew Pressley:
Elementary Differential geometry, Springer.
The last week of this course will cover material that is not discussed
in our textbook. You may wish to consult the first few pages of
these notes (postscript format).
Course outline (will be updated):
Parametrized and embedded curves: arc length, curvature and torsion,
Frenet formulas, four vertex theorem, isoperimetric inequality,
Crofton's formula
Surfaces in three dimensions: Surface patches, tangent spaces, normal
vectors, isometries and conformal maps, first and second fundamental
form, ruled surfaces, surfaces of revolution, minimal surfaces;
Notions of curvature: Gauss curvature, mean curvature, principal curvature, lines of curvature;
Weingarten matrix, Gauss map, Gauss and Codazzi-Mainardi equations,
theorema egregium,
Parallel transport: geodesics, covariant derivative, Christoffel symbols,
geodesic coordinates,
Gauss-Bonnet theorem, zeroes of vector fields, Poincare-Hopf theorem
Introduction to higher-dimensional manifolds
For some examples of famous curves, click
here.