Generalized de Sitter Space in $n$-dimensional Minkowski Space

In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and sufficient conditions are obtained for a hypersurface to be timelike, null, or spacelike in the generalized model; in the non-null case, the geometry is given by a warped product. Several examples of timelike, null, and spacelike hypersurfaces are presented. Lastly, we calculate the Ricci tensor and scalar curvature for a special family of 4-dimensional generalized de Sitter spaces.