Projectively Equivalent Riemannian Spaces as Quasi-bi-Hamiltonian Systems

The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton–Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannian space has a projectively equivalent metric its geodesic flow is a quasi-bi-Hamiltonian system. The existence of involutive first integrals of the geodesic flow, quadratic in the momenta, follows by a standard type of argument. When these integrals are independent they generate a Stäckel system.