On Killing vector fields on a tangent bundle with g-natural metric Part I

The tangent bundle of a Riemannian manifold (M, g) with a non-degenerate g− natural metric G that admits a Killing vector field is investigated. Using Taylor’s formula (TM,G) is decomposed into four classes. The equivalence of the existence of Killing vector field on M and TM is proved. More detailed investigaton of the classes is done in the part II of the paper