Quantum Integrability for the Beltrami-Laplace Operator for GeodesicallyEquivalent Metrics. Integrability Criterium for Geodesic Equivalence. Separation of Variables

摘要

Given two Riemannian metrics on a closed connected manifold M(sup n), the authorsconstruct self-adjoint differential operators such that if the metrics have the same geodesics then the operators commute with the Beltrami-Laplace operator of the first metric and pairwise commute. If the operators commute and if they are linearly independent, then the metrics have the same geodesics. If all eigenvalues of one metric with respect to the other are different at least at one point of the manifold the authors can globally separate the variable in the equation on eigenfunctions of the Beltrami-Laplace operator.