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Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; curvatures
In [7], it is proved that all $g$-natural metrics on tangent bundles of $m$-dimen\-sional 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.
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