Submanifolds In Pesudo-Riemannian Manifold

Research on submanifolds in pseudo-Riemannian manifold is an important field in differential geometry.Particularly,spacelike hypersurfaces’ related properties in Lorentz manifolds attract much attention from geometrists and physicians.If we give additional conditions on the basis of the condition that the hypersurfaces are spacelike,then we can obtain lots of conclutions.The paper introduces spacelike hypersurfaces in Lorentz manifolds,on the basis of relevant lemmas we prove an inequality.When the equality of the inequality holds,The spacelike hypersurface M has two different principal curvatures and their multiplicities are n-1 and 1.Then,on the basis of the inequality we get that the second fundamental form of the compact spacelike hypersurface M in R1n+1 satisfies an inequality.