The Space-like Hypersurfaces Of Locally Symmetric Lorentz Spaces

Based on the work of Choi [1,2], we introduce and study a new space M which satisfy the following conditions:(1) for any space-like vector u and any time-like vector v, K(u,v)=-c1/c2,(2) for any space-like vector u and v, K(u,v)≤c2.We call that M which satisfies condition (1) and (2) is a Lorentz space which satisfies condition (*).The purpose of this paper is to study compact space-like hypersurfaces with R aH+b in a locally symmetric Lorentz space. We investigate the relationship between n2H2and4(n-1)c by the self-adjoint operator Γ, then we get an estimate of the scalar curvature R.Theorem1Let N be a compact space-like hypersurface of dimensional n(n>2) with R=aH+b in a locally symmetric Lorentz space M which satisfies condition (*). then scalar curvature satisfies or where c=2c2+c1/c2.Meanwhile, we also obtain a sufficient condition of N to be totally umbilical.Theorem2Let N be a compact space-like hypersurface of dimensional n (n>2) with R=aH+b in a locally symmetric Lorentz space M which satisfies condition (*), and or When then the following properties hold.(1) If a2≥4b, then H=1/2(-a±(?)) and M is totally umbilical; (2) M has two distinct principal curvatures one of which is simple if and only if na(n-2)H3+{a2(n-l)2+2(n-2)c+n(n-2)6}H2+2a(n-1){c+(n-1)b}H+{c+(n-1)b}2=0, where c=2c2+c1/c2.