The exponential map based at a singularity

We study isolated singularities of a space embedded in a smooth Riemannian manifold from a differential geometric point of view. While there is a considerable literature on bi-lipschitz invariants of singularities, we obtain a more precise (complete asymptotic) understanding of the metric properties of certain types of singularities. This involves the study of the family of geodesics emanating from the singular point. While for conical singularities this family of geodesics, and the exponential map defined by them, behaves much like in the smooth case, the situation is very different in the case of cuspidal singularities, where the exponential map may fail to be locally injective. We also study a mixed conical-cuspidal case. Our methods involve the description of the geodesic flow as a Hamiltonian system and its resolution by blow-ups in phase space. 

 

This is joint work with Vincent Grandjean.