On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Abstract

Let $$M^{n}$$ be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $$L_{p}^{n+p}$$ of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold $$M^{n}$$ be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold $$L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}$$, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.