Gap Theorems Of Spacelike Submanifolds In Locally Symmetric Pseudo-riemannian Manifolds

In this paper,we study the linear Weingarten spacelike submanifolds in lo-eally symmetric pseudo-Riemannian manifolds.By applying the elliptic operator L extended by Cheng-Yau to the mean curvature H,we get L(nH),and estimate L(nH)by the bound of the sectional curvature of the ambient space.And then using the maximum principle,we get that the mean curvature H is a constant,and conclude the gap theorems of linear Weingarten spacelike submanifolds in locally symmetry pseudo-Riemannian manifolds,or spacelike hypersurface with constant scalar curvature in Lorentz space forms.There are three parts in this paper1.We study the complete linear Weingarten spacelike submanifold in the locally symmetric pseudo-Riemannian manifold Np n+p.A,ssume that the sectional evrrvature of Npn-p be bounded,we estimate L(nH).Then applyilg the Hopf strong maximum principle on L(nH).we find L(nH)=0.which implies either Mn is a totally geodesic submanifold.or Mn is a isoparametric submanifold under the assumption that the squared length of the second fundamental form be bounded from below2.We study the complete linear Weingarten spacelike hypersurface in the local symmetric Lorentz space L1n+1.The research idea similar to that of Riemannian space from,applying Omori-Yan maximum principle to L(nH)for hypersurface Mn of L1n+1,thus the gap theorem of hypersurface Mn is obtained,at the same time.when the supremum of |Φ|2 is reached,we prove that the hypersurface has two principal curvatures.3.We study spacelike hypersurfaces with constant scalar curvature in the Lorentz space form.Using extended Okumura lemma,we estimate L(|Φ|2)and get the upper and lower bounds of |Φ|2 by applying Omori-Yau maximaum principle on T(|Φ|2).And then,we find Mn is an isoparametric hypersurface from Hopf strong maxi-maum principle.