# Compact space-like hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

## Yaning Wang and Ximin Liu

Address:

Department of Mathematics, South China University of Technology, Guangzhou 510641, GuangDong, P. R. China

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, LiaoNing, P. R. China

E-mail:

wyn051@163.com.cn

ximinliu@dlut.edu.cn

Abstract: A new class of $(n+1)$-dimensional Lorentz spaces of index $1$ is introduced which satisfies some geometric conditions and can be regarded as a generalization of Lorentz space form. Then, the compact space-like hypersurface with constant scalar curvature of this spaces is investigated and a gap theorem for the hypersurface is obtained.

AMSclassification: primary 53C42; secondary 53C15.

Keywords: space-like hypersurface, constant scalar curvature, second fundamental form, locally symmetric Lorentz space.

DOI: 10.5817/AM2012-3-163