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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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 
@article{CMUC_2004_45_4_a2,
     author = {Abbassi, Mohamed Tahar Kadaoui},
     title = {Note on the classification theorems of $g$-natural metrics on the tangent bundle of a {Riemannian} manifold $(M,g)$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {591--596},
     year = {2004},
     volume = {45},
     number = {4},
     mrnumber = {2103077},
     zbl = {1097.53013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2004_45_4_a2/}
}
TY  - JOUR
AU  - Abbassi, Mohamed Tahar Kadaoui
TI  - Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 2004
SP  - 591
EP  - 596
VL  - 45
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMUC_2004_45_4_a2/
LA  - en
ID  - CMUC_2004_45_4_a2
ER  - 
%0 Journal Article
%A Abbassi, Mohamed Tahar Kadaoui
%T Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$
%J Commentationes Mathematicae Universitatis Carolinae
%D 2004
%P 591-596
%V 45
%N 4
%U http://geodesic.mathdoc.fr/item/CMUC_2004_45_4_a2/
%G en
%F CMUC_2004_45_4_a2
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/