On the Geometry of Complete Spacelike LW-Submanifolds in Locally Symmetric Semi-Riemannian Spaces

TL;DR

This paper investigates the geometry of complete spacelike linear Weingarten submanifolds in semi-Riemannian spaces, establishing rigidity results and characterizations under various curvature and analytic conditions.

Contribution

It introduces new rigidity theorems for spacelike submanifolds using a Simons-type formula and the Cheng-Yau operator, generalizing existing classification results.

Findings

Submanifolds are either totally umbilical or isoparametric under certain conditions.

Established sharp inequalities relating second fundamental form and mean curvature gradient.

Derived classification theorems unifying previous results in semi-Riemannian geometry.

Abstract

Let $M^{n}$ be an $n$-dimensional complete spacelike linear Weingarten submanifold immersed in a locally symmetric semi-Riemannian space $\mathbb{L}_{q}^{n+p}$ of index $q$, with parallel normalized mean curvature vector field and flat normal bundle. Assuming that $M^{n}$ satisfies suitable curvature constraints, we investigate rigidity results for such submanifolds. By combining a Simons-type formula for spacelike submanifolds with analytic techniques involving the Cheng-Yau modified operator $\mathcal{L}$, we establish sharp inequalities relating the traceless second fundamental form and the gradient of the mean curvature. As applications, we obtain several characterization results showing that $M^{n}$ must be either totally umbilical or isoparametric. More precisely, we derive rigidity results under three distinct frameworks: via the Omori-Yau maximum principle, via the $\mathcal{L}$-parabolicity of the underlying manifold, and under an integrability condition on the gradient of the mean curvature. These results generalize and unify known classification theorems for spacelike submanifolds satisfying linear Weingarten relations in semi-Riemannian ambient spaces.