Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

In this paper, by supposing a natural comparison inequality on the positive r-th mean curvatures of the hypersurface, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces immersed in a semi-Riemannian warped product of constant sectional curvature. Generalizing the above results, under a restriction on the sectional curvature or the Ricci curvature tensor of the fiber of a warped product, we also prove some new rigidity theorems in semi-Riemannian warped products. Our main results extend some recent Bernstein-type theorems proved in [12, 13, 14].