Characterizing closed linear Weingarten spacelike submanifolds immersed in the de Sitter space
This paper is devoted to the study of space-like submanifolds immersed in the de Sitter space. The study of submanifold theory gave rise to some useful tools to address certain physical problems, such as spacetime singularities and gravitational collapse, but from a mathematical point of view the study of space-like submanifods immersed in a Lorentzian manifold is interesting because of their geometric properties. Recall that the \((n+p)\)-dimensional de Sitter space \(\mathbb{S}_p^{n+p}\) of index \(p\) is the pseudo-Riemannian manifold of constant sectional curvature 1 given by the hyperquadric \[ \mathbb{S}_p^{n+p}=\left\{x\in\mathbb{R}_p^{n+p+1}\colon \langle x,x\rangle_p=1\right\} \] of the \((n+p+1)\)-dimensional Lorentz-Minkowski space of index \(p\), i.e., the Euclidean space \(\mathbb{R}^{n+p+1}\) endowed with the metric \(\langle \cdot,\cdot\rangle_p = -dx_1^2-\dots-dx_p^2+dx_{p+1}^2+\dots + dx_{n+p+1}^2\). In Theorem 1, the authors obtain a sharp integral inequality involving the norm of the traceless second fundamental form (also known as the total umbilicity tensor) of a closed linear Weingarten space-like submanifold \(M^n\), \(n\geq2\), immersed with parallel normalized mean curvature vector field in the de Sitter space \(\mathbb{S}_p^{n+p}\) of index \(p\geq1\), where the equality holds if and only if \(M^n\) is totally umbilical. In Theorem 2, they give a sharp integral inequality for closed linear Weingarten space-like hypersurfaces satisfying an Okumura-type condition on the norm of the total umbilicity tensor. These two results give different integral inequalities and different ways to characterize the totally umbilical round spheres of the de Sitter space.
parallel normalized mean curvature vector field
closed linear Weingarten space-like submanifolds
scientific article; zbMATH DE number 7375689
scientific article