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Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime
Author(s) -
Lima Henrique Fernandes,
Velásquez Marco Antonio Lázaro
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200341
Subject(s) - mathematics , uniqueness , vector field , hypersurface , mathematical analysis , spacetime , conformal map , mean curvature , context (archaeology) , mathematical physics , norm (philosophy) , second fundamental form , constant (computer programming) , pure mathematics , killing vector field , order (exchange) , curvature , geometry , physics , paleontology , quantum mechanics , political science , computer science , law , biology , programming language , finance , economics
We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V . In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V , we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett [29][J. Jellett, 1853].