Geodesic
Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 591-596.

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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.
Classification : 53A55, 53C07, 58A32
Keywords: Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; curvatures 
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Abbassi, Mohamed Tahar Kadaoui. Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 591-596. http://geodesic.mathdoc.fr/item/CMUC_2004__45_4_a2/