We investigate the geometry of complete spacelike hypersurfaces (immersed) in a generalized Robertson-Walker spacetime \(-I\times _fM^{n}\). Under suitable constraints on the sectional curvature of the Riemannian fiber \(M^n\), on the warping function f and on the future mean curvature (that is, the mean curvature function with respect to the future-pointing Gauss map of the spacelike hypersurface), we are able to prove that such a spacelike hypersurface must be a slice \(\{t\}\times M^{n}\) of the ambient spacetime. Nonexistence and Calabi–Bernstein type results concerning entire spacelike graphs constructed over the Riemannian fiber \(M^n\) are also obtained, as well as applications to the Einstein–de Sitter and steady state type spacetimes are given. Our approach is based on the so-called Omori–Yau’s generalized maximum principle and on certain integrability properties due to Yau.

我们研究了广义罗伯逊-沃克时空\（-I \ times _fM ^ {n} \）中完整的类空超曲面（浸没）的几何形状。在对黎曼纤维\（M ^ n \）的截面曲率的适当约束下，在翘曲函数f和未来平均曲率（即相对于像空的未来指向高斯图的平均曲率函数）上超曲面），我们能够证明这种类似空间的超曲面必须是环境时空的一个切片\（\ {t \} \ times M ^ {n} \）。关于存在于黎曼纤维\（M ^ n \）上的整个空间图的不存在和Calabi–Bernstein型结果还获得了Einstein-de Sitter的应用，并给出了稳态类型的时空。我们的方法基于所谓的Omori-Yau的广义最大原理和基于Yau的某些可积性。