Metrics of Kaluza-Klein type on the anti-de Sitter space H13

We introduce and study a new family of pseudo-Riemannian metrics on the anti-de Sitter three-space $H^3_1$. These metrics will be called “of Kaluza-Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle $T_1 H_2(κ)$. For any choice of three real parameters λ,μ, ν \neq 0, the pseudo-Riemannian manifold $(H^3_1, g_λμν)$ is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures (ϕ, ξ, η), compatible with $g_λμν$ , such that (ϕ, ξ, η, g_λμν) is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three-parameter family of homogeneous metric mixed 3-structures on the anti-de Sitter three-space.