Spacelike Hypersurfaces in de Sitter Space

Spacelike Hypersurfaces in de Sitter Space

A closed conformal vector field in de Sitter space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mn>1</mn></mrow><mrow><mi>...

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Bibliographic Details
Main Authors:	Yanlin Li, Mona Bin-Asfour, Kholoud Saad Albalawi, Mohammed Guediri
Format:	Article
Language:	English
Published:	MDPI AG 2025-02-01
Series:	Axioms
Subjects:	spacelike hypersurfaces de Sitter space Ricci curvature conformal vector field Robertson–Walker space
Online Access:	https://www.mdpi.com/2075-1680/14/3/155
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Summary:	A closed conformal vector field in de Sitter space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced separators="" open="(" close=")"><mover><mi>c</mi><mo>¯</mo></mover></mfenced></mrow></semantics></math></inline-formula> induces a vector field on a spacelike hypersurface <i>M</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced separators="" open="(" close=")"><mover><mi>c</mi><mo>¯</mo></mover></mfenced></mrow></semantics></math></inline-formula>, referred to as the induced vector field on <i>M</i>. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced separators="" open="(" close=")"><mover><mi>c</mi><mo>¯</mo></mover></mfenced></mrow></semantics></math></inline-formula> is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces.
ISSN:	2075-1680