Linear Weingarten Spacelike Submanifolds In Semi-riemannian Spaces

In this artical,we mainly investigated the complete linear Weingarten space-like submanifolds Mn in locally symmetric semi-Riemannian space Qpn+p.By using the generalized maximum prinpicle and Cheng-Yau's modified operator L,we study the problems for Mn being totally geodesic and isoparametric.The primary results obtained in this paper include the following two parts.1.The complete linear Weingarten space-like submanifolds in locally symmetric semi-Riemannian space are studied.By using the generalized maximum prinpicle and the operator L,We disscuss the pinching problem on the square of the length of the second fundamental form S and obtain a rigidity theorem for Mn being totally geodesic.2.The complete linear Weingarten space-like hypersurfaces with two distinct principle curvatures in locally symmetric Lorentz space L1n+1 are studied.we prove that such hypersurfaces must be isoparametric according to the Hpof maximum principle,and meanwhile a pinching conclusion about.sup|?| is obtained,where|?|2 = S-nH2.