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The tangent bundle of a Riemannian manifold (M,g) with non-degeneratedg-natural metric G that admits a Killing vector field is investigated. UsingTaylor's formula (TM,G) is decomposed into four classes that are investigatedseparately. The equivalence of the existence of Killing vector field on M andTM is proved.  Key words: Riemannian manifold, tangent bundle, g-natural metric, Killingvector field, non-degenerate metric.

On a classification of Killing vector fields on a tangent bundle with g-natural metric