On isotropic submanifolds and evolution of quasicaustics

Abstract

We study classification problems for generic isotropic submanifolds. The classification list of simple and unimodal singularities is obtained and the generic evolutions of quasicaustics in small dimension are classified. Examples encountered in geometric optics are presented. 0. Introduction and preliminaries. Let X be a manifold, and ω be a 2-form on X. The pair (X, ω) is called a symplectic manifold if ω is closed, i.e. dω = 0 and nondegenerate [AM]. The representative model of a symplectic manifold is a cotangent bundle T*M, endowed with the canonical 2-form CUM = dϋu, where the 1-form $M on T*M (Liouville form) is defined by (u, ϋM) = (TπM(u), τ r M (u)) , for each u e TT*M. The mapping TUM is the tangent mapping of %: T*M -» M and ττ*M: TT*M —• T*M is the tangent bundle projection. If (#,•) are local coordinates introduced in M, and (p*, q{) are corresponding local coordinates in T*M then O>M has the normal (Darboux) form coM = EU<tPi*dqi [We].