We consider a one-parameter family of new extrinsic differential geometries on hypersurfaces in hyperbolic space. Recently, the second author and his collaborators have constructed a new geometry which is called horospherical geometry on hyperbolic space. There is another geometry which is the famous Gauss–Bolyai–Robechevski geometry (i.e., the hyperbolic geometry) on hyperbolic space. The slant geometry is a one-parameter family of geometries which connect these two geometries. Moreover, we construct a one-parameter family of geometries on spacelike hypersurfaces in de Sitter space.
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Mikuri Asayama, Shyuichi Izumiya, Aiko Tamaoki, Handan Yıldırım, Slant geometry of spacelike hypersurfaces in hyperbolic space and de Sitter space. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 371–400DOI 10.4171/RMI/681