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Let (M,g) be a Riemannian manifold and G a nondegenerate g-natural metric onits tangent bundle T M . In this paper we establish a relation between theJacobi operators of (M,g) and that of (T M,G). In the case of a Riemanniansurface (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 anOsserman manifold.

g-natural metrics on tangent bundles and Jacobi operators