Systems of rays in the presence of distribution of hyperplanes

Abstract

Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems vH on T X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V ) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields vH are horizontal with respect to V . As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered. Introduction. Let X be a differentiable manifold. Let ω be a differential 1-form on X, and V a distribution of tangent hyperplanes in T X annihilated by ω. We consider the Hamiltonian systems vH , with Hamiltonian H on T X, which are horizontal with respect to V , i.e. the projections of the bicharacteristics of vH onto the base of T X are tangent to V . We introduce the notion of the space of geodesics-rays as the reduced symplectic space M of bicharacteristics on H−1( 12 ) equipped with the reduced symplectic form μ defined by the relation ωX |H−1( 2 ) −π μ = 0, where ωX is the canonical Liouville form on T X and π is the projection onto M along bicharacteristics (cf. [9]). Any Lagrangian submanifold L of (M,μ) is called a system of rays. Its counterimage π−1(L) represents an optical system of rays in the phase space (T X,ωX) of geometric optics. The graph of π is a Lagrangian submanifold in the product symplectic structure (T X ×M,π 2μ − π 1ωX), where πi are the projections of the cartesian product T X×M . Then we have a generating function G for graphπ (cf. [6]), which helps to determine the structure of the counterimage π−1(L). Now the stability notion 1991 Mathematics Subject Classification: Primary 57R45, 53B21; Secondary 58F05.