A class of metrics on tangent bundles of pseudo-Riemannian manifolds

H. M. Dida and A. Ikemakhen

University Moulay Tahar, Faculty of the Sciences and Technologies, Saïda, Algeria
University Cadi-Ayyad, Faculty of the Sciences and Techniques, Marrakech, Morocco

E-mail: ikemakhen@fstg-marrakech.ac.ma

Abstract: We provide the tangent bundle $TM$ of pseudo-Riemannian manifold $(M,g)$ with the Sasaki metric $g^s$ and the neutral metric $g^n$. First we show that the holonomy group $H^s$ of $(TM ,g^s)$ contains the one of $(M,g)$. What allows us to show that if $(TM ,g^s)$ is indecomposable reducible, then the basis manifold $(M,g)$ is also indecomposable-reducible. We determine completely the holonomy group of $(TM ,g^n)$ according to the one of $(M,g)$. Secondly we found conditions on the base manifold under which $(TM ,g^s)$ ( respectively $(TM ,g^n)$ ) is Kählerian, locally symmetric or Einstein manifolds. $(TM ,g^n)$ is always reducible. We show that it is indecomposable if $(M,g)$ is irreducible.

AMSclassification: primary 53B30; secondary 53C07, 53C29, 53C50.

Keywords: pseudo-Riemannian manifold, tangent bundle, Sasaki metric, neutral metric, holonomy group, indecomposable-reducible manifold, Einstein manifold.