HCRP project 2008
HCRP 2008 Project
Conchoids in Differential Geometry
Office: SciCtr 434
This HCRP (Harvard College Research Program)
project took place in the second half of the Fall 2008 semester. Michael Teodorescu studied with me the
exponential map in differential geometry. It allows us to explore some elementary Riemannian
geometry and calculus of variations. Michael's motivation are biological predator-pray paths
in various metric setups, Oliver's interest is the geometry of wave fronts
and caustics. We made experiments with the exponential map in the simplest cases,
especially for compact perturbations of the flat and hyperbolic plane. Given a point P, we can evolve
a curve along the geodesic flow starting at P. The evolved curve is called a conchoid of the
original curve. In geodesic polar coordinates, the evolution is explicitly given by
rs(t) = r(t) + s at least for a short time s.
In the flat plane, the hyperbolic space and the sphere, the conchoid evolution can be
written down in closed form for all times.
While the geodesic flow on a general Riemannian plane is a complicated dynamical system in general,
for compact perturbations of the flat or hyperbolic metric, the geodesic flow becomes a
scattering problem for which caustics are sets which can be computed when sufficiently close
to the uniform flat or hyperbolic case.