Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz–Minkowski space

In this paper we define one-parameter families of Legendrian double fibrations in the products of pseudo-spheres in Lorentz–Minkowski space which are the extensions of four Legendrian double fibrations in the previous research (Izumiya, 2009 [9]). We show that these are contact diffeomorphic to each other. Moreover, we construct one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in these pseudo-spheres as applications of such extensions of the Legendrian double fibrations.