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Complete linear Weingarten spacelike submanifolds immersed in the anti-de Sitter space

Volume 165 / 2021

Henrique Fernandes de Lima Colloquium Mathematicum 165 (2021), 117-130 MSC: Primary 53C42; Secondary 53A10, 53C20, 53C50. DOI: 10.4064/cm7941-5-2020 Published online: 14 December 2020


We deal with $n$-dimensional complete linear Weingarten spacelike submanifolds having nonnegative sectional curvature and immersed in the anti-de Sitter space $\mathbb H_p^{n+p}$ of index $p$ with parallel normalized mean curvature vector field. We recall that a spacelike submanifold is said to be linear Weingarten when its mean and normalized scalar curvature functions are linearly related. We prove that under suitable constraints on the mean curvature function, such a spacelike submanifold must be either totally umbilical or isometric to a product $M_1\times \cdots \times M_k$, where the factors $M_i$ are totally umbilical submanifolds of $\mathbb H_p^{n+p}$ which are mutually perpendicular along their intersections. Furthermore, when this spacelike submanifold is assumed to be compact (without boundary) with positive sectional curvature, we also obtain a rigidity result.


  • Henrique Fernandes de LimaDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.429-970 Campina Grande, Paraíba, Brazil

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