Focusing geodesics - geometry and topology

Thomas Waters, University of Portsmouth. Part of geometry seminar series.

Geodesics are the straight lines of curved surfaces. If we consider a spray of geodesics emanating from a certain point then, due to the curvature of the surface, the geodesics can focus along curves. These focusing curves are variously called caustics, envelopes or the conjugate locus, and they can be terrifically complex. In particular they have spikes or cusps, and they can fold up on themselves. We will show there is a simple relationship between the number of cusps and the rotation index of the conjugate locus on convex surfaces, and how we can use this to then prove other strong results. Finally we will consider the extension to focusing of geodesics in 3-dimensional manifolds and some early results.