EVENT DETAILS AND ABSTRACT

Title: Geodesics at singular points of generic cuspidal surfaces
Speaker: Vincent Grandjean
Speaker Info: Fields Institute
Brief Description:
Special Note:
Abstract:

Assume a Riemannian manifold (M,g) is given. Let X be a locally closed subset of M, that is singular at some of its point, that is X is not a submanifold at this point. We can think of singular real algebraic sets, or germs or real analytic sets as a model of the singularities we are interested in dealing with. The smooth part of X comes equipped with a Riemannian metric induced from the ambient one. We would like to understand how do geodesics on the regular part of X behave in a neighbourhood of a singular point. It turns out that very little is known (or even explored) about very elementary singularities (conical, edges or corners). One of the purpose of specifying these simple singular "manifolds" was to study the propagation of singularities for the wave equation on such a singular "manifold" (Melrose, Vasy, Wunsch,...). In this joint work with Daniel Grieser (Oldenburg, Germany), we want to address a naive and elementary question: Given an isolated surface singularity X in (M,g), can a neighbourhood of the singular point be foliated by geodesics reaching the singular point ? Then could we define an exponential-like mapping at a such point ? This property is true for conical singularities of any dimension (Melrose & Wunsch). With D. Grieser, we have exhibited very simple examples of non-conical real surfaces with an isolated singularity, and cuspidal-like, in a 3-manifold, such that the geodesics reaching the singular point behave differently according to the considered class.