A new approach to the study of spacelike submanifolds in a spherical Friedmann–Lemaître–Robertson–Walker spacetime: characterization of the stationary spacelike submanifolds as an application

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 56; no. 24; pp. 245202 - 245216
• Main Authors: Ferreira, D, Lima, E A, Palomo, F J, Romero, A
• Format: Journal Article
• Language: English
• Published: IOP Publishing 16.06.2023
• Subjects: isometric embedding; Lorentz–Minkowski spacetime; mean curvature; Robertson–Walker spacetime; spacelike submanifolds
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/acd502

Abstract A natural codimension one isometric embedding of each ( n + 1 ) -dimensional spherical Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime I × f S n in the ( n + 2 ) -dimensional Lorentz–Minkowski spacetime L n + 2 permits to contemplate I × f S n as a rotation Lorentzian hypersurface in L n + 2 . After a detailed study of such Lorentzian hypersurfaces, any k -dimensional spacelike submanifold of such an FLRW spacetime can be contemplated as a spacelike submanifold of L n + 2 . Then, we use that situation to study k -dimensional stationary (i.e. of zero mean curvature vector field) spacelike submanifolds of the FLRW spacetime. In particular, we prove a wide extension of the Lorentzian version of the classical Takahashi theorem, giving a characterization of stationary spacelike submanifolds of I × f S n when contemplating them as spacelike submanifolds of L n + 2 .