Linear Weingarten Spacelike Submanifolds in Semi-Riemannian Space Form

We consider the linear Weingarten spacelike submanifold immersed in semi-Riemannian space form $\mathbb {N}^{n+p} _q (c)$ of constant sectional curvature $c$ and index $q$, where $p \ge q>1$. In this paper, our approach involves an appropriately generalized maximum principle for a suitable Cheng–Yau modified operator so as to obtain that the submanifold is definitely either totally umbilical or isometric to a product space.