Hokkaido University Preprint Series in Mathematics
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-Boryay-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 oneparameter family of geometries on spacelike hypersurfaces in de Sitter space.