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Keywords:
locally symmetric semi-Riemannian space; mean curvature vector field; complete linear Weingarten spacelike submanifold; totally umbilical submanifold; isoparametric submanifold; $\mathcal L$-parabolicity
locally symmetric semi-Riemannian space; mean curvature vector field; complete linear Weingarten spacelike submanifold; totally umbilical submanifold; isoparametric submanifold; $\mathcal L$-parabolicity
Summary:
In this paper, we deal with $n$-dimensional complete linear Weingarten spacelike submanifolds 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 $p>1$, which obeys some curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). Under appropriate hypothesis, we are able to prove that such a spacelike submanifold is either totally umbilical or isometric to an isoparametric submanifold of the ambient space. For this, we use three main core analytical tools: a suitable version of the Omori--Yau maximum principle, parabolicity with respect to a modified Cheng--Yau operator and a certain integrability property.
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