Characterizing closed linear Weingarten spacelike submanifolds immersed in the de Sitter space
type: Item
instance of: scholarly article
title: Characterizing closed linear Weingarten spacelike submanifolds immersed in the de Sitter space
zbMATH Open document ID: 1475.53059
author: Henrique Fernandes de Lima
author: Eudes L. De Lima
published in: São Paulo Journal of Mathematical Sciences
publication date: 2021-07-28T00:00:00Z
review text: 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.
reviewed by: M. Ferreiro-Subrido
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 7375689
zbMATH Keywords: de Sitter space
zbMATH Keywords: parallel normalized mean curvature vector field
zbMATH Keywords: closed linear Weingarten space-like submanifolds
zbMATH Keywords: sharp integral formula
zbMATH Keywords: totally umbilical round spheres
zbMATH Keywords: Okumura type condition
MaRDI profile type: Publication
Wikidata QID: Q107169125
full work available at URL: s40863-021-00229-0
OpenAlex ID: W3141281193
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DOI: 10.1007/S40863-021-00229-0
description: scientific article; zbMATH DE number 7375689