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...

Let E Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝm-n. As an application we give some estimates of card E Aff(Γ,G, m).

This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.