- Elementary Differential Geometry
(second edition), Barrett O'Neill.
The official prerequisite is a 2xx-level calculus course; we will also
require a certain amount of mathematical maturity.
- There will be weekly homework
assignments, due on Wednesdays, often with problems assigned
from the textbook. The homework counts for 30% of the course grade.
- There will be two hour-tests on Fridays in class.
These will be on Feb 18 and Mar 31; the exact material covered
on each will be announced later. These tests count for 40% of your grade.
The final exam covers the entire course, and counts for 30% of the course
grade. The final exam will be 7-10pm Tue 9 May.
This course covers the differential geometry of smooth curves and surfaces
in ordinary three-dimensional space. The basic approach is to look for
those properties (the curvatures) of the shape which are independent
of how we parameterize the curve or surface, and of where it is placed
in space. For a surface, there is one particular combination of
curvatures, the Gauss curvature, which is in fact intrinsic. This means
that if we bend the surface without stretching it (like rolling a piece
of paper) this curvature is unchanged. The course will end with the
surprising Gauss-Bonnet theorem, which says that furthermore
the integral of the Gauß curvature over the whole surface is a
topological invariant, unchanged even under stretching.