Abstract:In [7], it is proved that all $g$-natural metrics on tangent bundles of $m$-dimensional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on $m$ 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 the $g$-natural metrics are, in fact, smooth on the set of all nonnegative real numbers.
Keywords: Riemannian manifold, tangent bundle, natural operation, $g$-natural metric, curvatures
AMS Subject Classification: Primary 53C07; Secondary 53A55