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Mohamed Tahar Kadaoui Abbassi

Note on the classication theorems of

g

-natural metrics on the tangent bundle of a Riemannian manifold

(M, g)

Comment.Math.Univ.Carolinae 45,4 (2004) 591-596.

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 origi- nally 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

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