  Let (M,g) be a Riemannian manifold and G a nondegenerate g-natural metric on
its tangent bundle T M . In this paper we establish a relation between the
Jacobi operators of (M,g) and that of (T M,G). In the case of a Riemannian
surface (M,g), we compute explicitly the spectrum of some Jacobi operators of
(TM,G) and give necessary and sufficient conditions for (T M,G) to be an
Osserman manifold.
