Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds

If a homogeneous space $G/H$ is acted properly discontinuously upon by a subgroup $\varGamma $ of $G$ via the left action, the quotient space $\varGamma \backslash G/H$ is called a Clifford–Klein form. In Calabi and Markus (Ann Math (2) 75: 63–76, 1962) proved that there is no infinite subgroup of the Lorentz group $O(n+1,\,1)$ whose left action on the de Sitter space $O(n+1,\,1)/O(n,\,1)$ is properly discontinuous. It follows that a compact Clifford–Klein form of the de Sitter space never exists. In the present paper, we provide a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous by using the techniques of differential geometry.