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Vincent Grandjean,
Fields Institute

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). The purpose of specifying such a singular set 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 can we define an exponential mapping at a such point ? This property is true for conical singularities of any dimension. 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.