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