Address.
M. T. H. K. Abbassi, Departement des
Mathematiques, Faculte des sciences Dhar El Mahraz,
Universite Sidi Mohamed Ben Abdallah,
B.P. 1796, Fes-Atlas, Fes, Morocco
M. Sarih,
Departement des Mathematiques et
Informatique, Faculte des sciences et techniques de
Settat,
Universite Hassan 1
B.P. 577, 26000 Morocco
E-mail. mtk_abbassi@Yahoo.fr
Abstract.
There is a class of metrics on the tangent bundle $TM$ of a
Riemannian manifold $(M,g)$ (oriented , or non-oriented,
respectively), which are 'naturally constructed' from
the base metric $g$ [Kow-Sek1]. We call them
``$g$-natural metrics" on $TM$. To our knowledge, the geometric
properties of these general metrics have not been studied yet.
In this paper, generalizing a process of Musso-Tricerri (cf.
[Mus-Tri]) of finding Riemannian metrics on $TM$ from
some quadratic forms on $OM \times \mathbb{R}^m$ to find
metrics (not necessary Riemannian) on $TM$, we prove that all
$g$-natural metrics on $TM$ can be obtained by Musso-Tricerri's
generalized scheme. We calculate also the Levi-Civita
connection of Riemannian $g$-natural metrics on $TM$. As
application, we sort out all Riemannian $g$-natural metrics
with the following properties, respectively:
1) The fibers of $TM$ are totally geodesic.
2) The geodesic flow on $TM$ is incompressible.
We shall limit ourselves to the non-oriented situation.
AMSclassification. Primary 53B20, 53C07, 53D25; Secondary 53A55, 53C24, 53C25, 53C50.
Keywords. Riemannian manifold, tangent bundle, natural operation, $g$-natural metric, Geodesic flow, incompressibility.