A classification of natural transformations transforming vector fields on -manifolds into affinors on the extended -th order tangent bundle over -manifolds is given, provided .
In [7], it is proved that all -natural metrics on tangent bundles of -dimensional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on and on the assumption that the base manifold is oriented, or non-oriented, respectively. The result was originally stated in [8] for the oriented case, but the smoothness was assumed and not explicitly proved. In this note, we shall prove that, both in the oriented and non-oriented cases, the functions generating...