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A Theorem on the First Eigencalue of the Laplacian for Compact Submanifolds of Euclidean Space

YANG Hui-Zhang1,Guo Shun-Zi2(1.Department of Mathematics,Hong He Teacher's College,Yunnan Mengzi 661100,China;2.Department of Mathenatice,Zhang Zhou Teacher's College,Zhangzhou 363000,China)  
In this paper,we give a global theorem:let x:Mn→Rn+p is immersion of a compact manifold M into Euclidean space R.let F(q) is the maxinum of the square of the norm of the second fundamental form along unit normal vector u,λ1 is the first eigencalue of Laplacian operator △,then we have equality if and only if x immerses M as a hypersphere in an(n+1) —dimensional linear sub-space of Rn+p.So we generalize R.C.Reillys result in[1].
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