The Spacelike Submanifolds In De Sitter Space

At first, we obtain an intrinsic inequality of the Ricci curvature tensor and scalar curvature for the spacelike submanifolds Mn in the de Sitter space Spn+p(c). In this way, we generalize a result for hypersurfaces due to H-D Pang, S-L Xu and D Shu [9] as followsTheorem1Let Mn be a spacelike submanifold immersed in the de Sitter space Spn+p(c). Denoted the Ricci curvature tensor and the scalar curvature of Mn by Ric and R respectively, we have the equality holds if and only if Mn is an Einstein manifold with constant scalar curvature cn(n—1).Secondly, we discuss the Yang-Mills fields of submanifolds in the sphere Sn+P. we obtain a theorem about the flat Yang-Mills connection over Mn.Theorem2let Mn be an n-dimensional compact submanifold in the sphere Sn+P and R▽be the Yang-Mills fields. We denote the mean curvature, the square length of the second fundamental form and the square length of the second funda-mental form in the mean curvature direction by H,σ,σH respectively, and we set φ=σ+{(σ-nH2)(σ-(3n-2)H2+2(n-2)|H|(σH-nH2)1/2+(n-2)2H2(σH-nH2)+n(n-1)2H4}1/2.If the Lp-norm of φ satisfies then R▽=0, where a=supM{R▽|,k1and k2are the constants in sobolev lemma (see Lemma1, p.6).