(This talk is based on a joint work with Mikuri Asayama, Shyuichi Izumiya and Aiko Tamaoki.)

It is known that there are three kinds of pseudo-spheres in Lorentz-Minkowski space which are called Hyperbolic space, de Sitter space and lightcone. A basic duality theorem for four Legendrian dualities related with these pseudo-spheres was obtained in [3]. These Legendrian dualities were extended in [4] depending on a parameter ϕ in[0, π/2]. Moreover, as an application of these extended Legendrian dualities, one-parameter families of extrinsic differential geometries on spacelike hypersurfaces in de Sitter space were constructed in [2]. We call these geometries which include the results of [5] as a special case slant geometry of spacelike hypersurfaces in de Sitter space. In this talk, we first mention about the basic framework of slant geometry of spacelike hypersurfaces in de Sitter space. Then, we review the notion of spacelike de Sitter Monge form which was introduced in [5]. Finally, we give some examples of spacelike hypersurfaces in de Sitter space by means of spacelike de Sitter Monge forms in sense of slant geometry.

References

[1] V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. I, Birkhauser  (1985).

[2] M. Asayama, S. Izumiya, A. Tamaoki and H. Yıldırım, Slant geometry of spacelike hypersurfaces in Hyperbolic space and de Sitter space, Revista Matematica Iberoamericana, 28(2) (2012), 371-400.

[3] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Moscow Mathematical Journal, 9 (2009), 325-357.

[4] S. Izumiya and H. Yıldırım, Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space, Topology and its Applications, 159 (2012), 509-518.

[5] M. Kasedou, Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space, Journal of Geometry, 94 (2009), 107-121.