Geodesic
On natural metrics on tangent bundles of Riemannian manifolds
Archivum mathematicum, Tome 41 (2005) no. 1, pp. 71-92.

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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.
Classification : 53A55, 53B20, 53C07, 53C20, 53D25
Keywords: Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; Geodesic flow; incompressibility 
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     author = {Abbassi, Mohamed Tahar Kadaoui and Sarih, Ma\^ati},
     title = {On natural metrics on tangent bundles of {Riemannian} manifolds},
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     pages = {71--92},
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     zbl = {1114.53015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2005__41_1_a6/}
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Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti. On natural metrics on tangent bundles of Riemannian manifolds. Archivum mathematicum, Tome 41 (2005) no. 1, pp. 71-92. http://geodesic.mathdoc.fr/item/ARM_2005__41_1_a6/