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

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...