On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form (M¯,g¯), where M¯=R×fM and g¯=ϵdt2+f2(t)gM. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where (M¯,g¯) is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ∂t and the so-called support function θ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below.