## Weiller F. C. Barboza, Henrique F. de Lima, Marco A. L. Velásquez

*Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space*

Comment.Math.Univ.Carolin. 64,1 (2023) 39-61.**Abstract: **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.

**Keywords:** locally symmetric semi-Riemannian space; mean curvature vector field; complete linear Weingarten spacelike submanifold; totally umbilical submanifold; isoparametric submanifold; \mathcal L-parabolicity

**DOI:** DOI 10.14712/1213-7243.2023.013

**AMS Subject Classification:** 53C42 53C21 53C50

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