# Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying L_k(x)=Ax+b

Document Type : Research Paper

Authors

1 Tarbiat Modares University, Iran

2 Tarbiat Modares University

Abstract

We study connected
orientable spacelike hypersurfaces
$x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the
Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and
index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~
where $L_k$ is the $textit{linearized operator}$ of
the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a
fixed integer $0leq k<n$, $A$ is a constant matrix and $b$ is a
constant vector.

We show that the only hypersurfaces satisfying that condition are
hypersurfaces with zero $H_{k+1}$ and constant $H_k$ ( when $cneq 0$ ), open pieces of totally umbilic hypersurfaces and open pieces
of the standard Riemannian product of two totally umbilic
hypersurfaces.

Keywords

Main Subjects

### History

• Receive Date: 30 May 2011
• Revise Date: 07 October 2011
• Accept Date: 07 October 2011