Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics

Abstract

Let two Riemannian metrics g and g on one manifold M ⁿ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian Δ _g of the metric g is one of these operators. For any x ∈ M ⁿ, consider the linear transformation G of T _x M ⁿ given by the tensor g ^Iαg_αj . If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation Δ _g f = μf on this torus.