Existence results for boundary value problems for fourth-order differential inclusions with nonconvex valued right hand side

Mohamed Tahar Kadaoui Abbassi and Maati Sarih

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

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.