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In this paper, we determine a class of possible restricted holonomy groups for a non-irreducible indecomposable pseudoriemannian manifold with signature (2,2 + n). In particular, we deduce that which associated to symmetric spaces; and give some examples of such spaces. Finally, we construct some examples of metrics whose restricted holonomy groups are not closed.

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...