On Isotropic Submanifolds and Evolution of Quasicaustics

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]. We recall that a submanifold C c {X, ω) is coisotropic if, at each x E C, the symplectic polar of TXC defined by C£ = {v e TXX: (v Λ u, ω) = 0 for every w e ΓXC} is contained in Γ X C By (vAu,ω) we denote the evaluation of ω on the pair of vectors υ, u e TXX. If C£ = ΓXC for each x G C then C is called the Lagrangian submanifold of X . In this case ω\c = 0, and dimC = ^dimX. We see that dimC^= codimC and {C^} forms the characteristic distribution of ω\c Thus the distribution D = \Jχec €χ * i^volutive. Maximal connected integral manifolds of D are called bicharacteristics. They form the characteristic foliation of C (cf. [AM]). D represents the generalized Hamiltonian system