Conchoids in Differential Geometry

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 r_s(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.