Equivalence of isotropic submanifolds and symmetry

The genesis of this paper lies in theoretical questions in geometrical diffraction theory where a central role is played by the systems of rays passing through a boundary of an obstacle ~aperture! ~cf. Refs. 1,2!. It is explained in Refs. 1,3,4 why the proper isotropic submanifolds of cotangent boundles ~phase spaces! do occur in geometrical diffraction and why the symmetry group of these objects appear as a natural feature of existing optical systems ~cf. Refs. 5,6!. Let F:R3R3X→R , F(x ,y ,a ,b ,q1 ,q2 ,q2) be the optical distance function from the wave front $(x ,y ,z):z5f(x ,y), f~0!50, f8~0!50% in the presence of an aperture parametrized by $(a ,b)PR: f (a ,b)>0% to the configurational point (q1 ,q2 ,q3)PX . If the incident ray goes from the point „x ,y ,f(x ,y)... to the point of an edge $ f (a ,b)50% of the aperture, then the diffracted rays form a cone in X ~cf. Refs. 2,6!. The natural subsystems of diffracted rays form those rays that are straight continuations of the incident rays. The system of incident rays passing through an edge of the aperture form an isotropic two-dimensional submanifold of T*X . This submanifold is described by the following equations: