%0 Journal Article
%T Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes
%A Junhong Dong
%A Ximin Liu
%J ISRN Geometry
%D 2014
%R 10.1155/2014/974695
%X We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle. 1. Introduction The mathematical interests for the study of spacelike hypersurfaces in spacetimes began in the seventies with the works of Cheng and Yau [1], Brill and Flaherty [2], Choquet-Bruhat [3], and later on with some other authors, such as in [4¨C6]. Moreover, the study of such hypersurfaces is also of interest from a physical point of view, because of its relation to several problems in general relativity. More recently, there has been an increasing interest in the study of spacelike hypersurfaces with constant higher order mean curvature, such as in [7¨C9]. At the same time, in [10, 11] there are system research works about the lightlike hypersurface of semi-Riemannian manifolds. In this paper, based on the previous result, we want to make an attempt to study the lightlike hypersurfaces of spacetimes. First of all, we are interested in the study of lightlike hypersurfaces in conformally stationary spacetimes, and the screen bundle with constant higher order mean curvature. First, we use the Newton transformations as the main analytical tool and study the Minkowski-type integral formulas of the lightlike hypersurface. The use of these kinds of formulas in the Lorentzian manifold was first started by Montiel in [12] for the spacelike hypersurface with constant mean curvature in de Sitter spacetimes, and it was continued by Al¨ªas et al. [13, 14] for more general spacetimes. Higher-order Minkowski formula for hypersurface was first obtained by Hsiung in [15] in Euclidean space, and by Bivens in [16] in the Euclidean sphere and hyperbolic space. In this paper, we obtain two Minkowski-type integral formulas about the higher-order mean curvature of the lightlike hypersurface as follows, which are called Minkowski formulas I and II. The Minkowski Formula I. Let be a compact lightlike hypersurface of a conformally stationary spacetime of constant sectional curvature, on which the screen bundle is integrable and Ricci tensor of the induced connection is symmetric; then the following conclusion holds: The Minkowski Formula II. Let be a compact lightlike hypersurface immersed into a conformally stationary
%U http://www.hindawi.com/journals/isrn.geometry/2014/974695/