Local intersections of Lagrangian manifolds correspond to catastrophe theory

Abstract

Two smooth map germs are right-equivalent if and only if they generate two Lagrangian submanifolds in a cotangent bundle which have the same contact with the zero-section. In this paper we provide a reverse direction to this classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a symplectic manifold have the same contact with a third Lagrangian submanifold if and only if the intersection problems correspond to stably right equivalent map germs. We, therefore, obtain a correspondence between local Lagrangian intersection problems and catastrophe theory while the classical version only captures tangential intersections. The correspondence is defined independently of any Lagrangian fibration of the ambient symplectic manifold, in contrast to other classical results. Moreover, we provide an extension of the correspondence to families of local Lagrangian intersection problems. This gives rise to a framework which allows a natural transportation of the notions of catastrophe theory such as stability, unfolding and (uni-)versality to the geometric setting such that we obtain a classification of families of local Lagrangian intersection problems. An application is the classification of Lagrangian boundary value problems for symplectic maps.