Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

In the phase space over a Riemann manifold we consider a submanifold 4 invariant with respect to a Hamilton flow, isotropic (i.e., the formpdq_Λis closed), and stable with respect to the first variation equation. For semiclassical wave- functions of the quantum Hamiltonian we propose a very simple global ansatz with an oscillation front onΛ. This ansatz has a form of an integral alongΛfrom Gaussian packets framed by an "amplitude." The amplitude is a parallel section of a bundle of polynomials. The connection on this bundle is generated by a symplectic connection with zero curvature on the normal symplectic bundle overΛ. The coefficients of such a connection are calculated explicitly in terms of infinitesimal symmetries, and also in adiabatic approximation. We investigate topological and geometrical objects arising as corrections to the Poincaré-Cartan invariant in quantization rule onΛand calculate spectral series for the quantum Hamiltonian.