Geodesic
A class of metrics on tangent bundles of pseudo-Riemannian manifolds
Archivum mathematicum, Tome 47 (2011) no. 4, pp. 293-308.

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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.
Classification : 53B30, 53C07, 53C29, 53C50
Keywords: pseudo-Riemannian manifold; tangent bundle; Sasaki metric; neutral metric; holonomy group; indecomposable-reducible manifold; Einstein manifold 
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     author = {Dida, H. M. and Ikemakhen, A.},
     title = {A class of metrics on tangent bundles of {pseudo-Riemannian} manifolds},
     journal = {Archivum mathematicum},
     pages = {293--308},
     publisher = {mathdoc},
     volume = {47},
     number = {4},
     year = {2011},
     mrnumber = {2876951},
     zbl = {1249.53020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011__47_4_a5/}
}
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Dida, H. M.; Ikemakhen, A. A class of metrics on tangent bundles of pseudo-Riemannian manifolds. Archivum mathematicum, Tome 47 (2011) no. 4, pp. 293-308. http://geodesic.mathdoc.fr/item/ARM_2011__47_4_a5/