Reticular Lagrangian singularities (Q1382028)

The author investigates a general situation where a hypersurface in a Riemannian manifold has an \(r\)-corner. In this case incident rays from each edge of the hypersurface to conormal directions yield a regular \(r\)-cubic configuration at a point of the cotangent bundle. The stability of smooth regular \(r\)-cubic configurations and the classification of stable caustics is investigated. The notion of reticular Lagrangian maps is introduced as well as various notions of stability for these maps. The classification of stable caustics is reduced to the classifications of function germs and as a result stable caustics are classified for manifolds of dimension \(\leq 6\). Pictures are also drawn for stable caustics in manifolds of dimension \(\leq 4\).