Sample project topics Math 451 Spring 2009
Bear in mind: The quality of a project is directly related to its having substantive content
and a well-defined,narrow focus. The following are intended to be suggestive. Most topics
need a narrowing of focus. Computer graphics and experiments might help to illuminate some
of the ideas and results.
•At each point of a surface you can follow the directions of the principal curvatures. Doing
so produces two systems of orthogonal curves on the surface. Develop the basic theory of
these “lines of curvature.” You might consider the case of an ellipsoid as a means to work
out some of the interesting properties—such as their being the intersection of the ellipsoid
with families of orthogonal surfaces—with that arise.
Source: Hilbert and Cohn-Vossen, Geometry and the Imagination
•An umbilic point on a surface is one where the directional curvature is the same in all
directions—as on the plane or sphere. On an ellipsoid most points aren’t umbilic, but there
are some special places that are. Discuss how to find these points and their connection to
lines of curvature.
Source: Hilbert and Cohn-Vossen, Geometry and the Imagination
•A sphere has the property that it can’t be bent—that is, deformed in a way that doesn’t
distort distance? However, if you remove an arbitrarily small patch from the sphere, the
remaining surface is bendable. Investigate these phenomena and develop arguments for
these claims.
Source: Hilbert and Cohn-Vossen, Geometry and the Imagination
•An important idea in differential geometry concerns moving a tangent vector on a surface
Sin such a way that all the vectors that arise are tangent to Sand are parallel in to the
intrinsic geometry of the surface. (This does not mean parallel as vectors in R3. It might
be that none of the vectors tangent to Sat one point are parallel as vectors in R3to a
tangent vector at another point.) Investigate the notion of parallel transport of vectors
which supplies an intrinsically meaningful definition ‘parallel.’
Use the notion of parallel transport on the sphere to describe the behavior of a Foucault
pendulum.
Sources: Oprea, Differential Geometry and its Applications
Oprea, Geometry and the Foucault pendulum (Amer. Math. Monthly, June-July 1995)
•The cycloid has the remarkable property that if an object moves along the curve free of
friction under a constant force—such as gravity over a small distance, it takes the same
amount of time to reach the bottom regardless of the height at which it starts. Establish
that the cycloid is the “tautochrone” (meaning ”same time”).
Source: McCleary, Geometry from a Differential Viewpoint
