Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

Ximin Liu,Junhong Dong

doi:10.1155/2014/974695

Abstract

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.

Highlights

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 [4,5,6]
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 are interested in the study of lightlike hypersurfaces in conformally stationary spacetimes, and the screen bundle with constant higher order mean curvature

Summary

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 [4,5,6]. We are interested in the study of lightlike hypersurfaces in conformally stationary spacetimes, and the screen bundle with constant higher order mean curvature. Let (M, g, S(TM)) be a compact lightlike hypersurface of a conformally stationary spacetime M of constant sectional curvature, on which the screen bundle S(TM) is integrable and Ricci tensor of the induced connection ∇ is symmetric; the following conclusion holds:. Let (M, g, S(TM)) be a compact lightlike hypersurface immersed into a conformally stationary spacetime (M, g) of constant sectional curvature, where the screen bundle S(TM) is integrable and the Ricci tensor of induced connection ∇ is symmetric; .

Preliminaries

The Shape Operator and Newton Transformation

Minkowski Integral Formula for Lightlike Hypersurface in RW Spacetime

Umbility of the Compact Lightlike Hypersurface in RW Spacetime

Compact Lightlike Hypersurface with Elliptic Point