Metrics of Kaluza–Klein type on the anti‐de Sitter space

Abstract

We introduce and study a new family of pseudo‐Riemannian metrics g̃λμν on the anti‐de Sitter three‐space H13. 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 T1double-struckH2(κ). For any choice of three real parameters λ,μ,ν≠0, the pseudo‐Riemannian manifold ()double-struckH13,trueg̃λμν is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures (φ,ξ,η), compatible with g̃λμν, such that (φ,ξ,η,trueg̃λμν) 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.

Hopf magnetic curves in the anti-de Sitter space 𝔿13

“…We shall now present the hyperbolic counterpart of the Hopf fibration S 3 → S 2 (1/2). For further details, we may also refer to [10] and [12]. Let us consider R 4 2 , the four-dimensional pseudo-Euclidean space equipped with the pseudo-Riemannian flat metric…”

“…For this reason, in analogy with the Riemannian case, we call ξ the hyperbolic Hopf vector field. It is easily seen that ξ is a globally defined Killing vector field [12].…”

“…We shall now describe the anti-de Sitter space H 3 1 in terms of paraquaternions, referring to [12] for more details (see also [22]). Consider the algebra B of paraquaternionic numbers over R generated by {1, i, j, k}, where −i 2 = j 2 = 1 and k = i j = − ji (they are also known as Gödel quaternions or split quaternions in Theoretical Physics).…”

“…With respect to g, the vector field ξ is time-like, while U,V are space-like. If ∇ is the Levi-Civita connection of the metric g, we have (see also Section 4 in [12])…”

Untitled

“…Unlike the Sasaki metric which shows a very rigid behaviour, the large class of g-natural metrics provides examples for several different interesting geometric properties (cf. [3,5,[7][8][9]12,14,15,21] and references therein).…”