欧氏空间中具有常数量曲率的超曲面的刚性

Rigidity of Hypersurfaces with Constant Scalar Curvature in Euclidean Space

设x:Mn →Rn+m为紧致黎曼流形Mn到欧氏空间的等距浸入.对于欧氏空间中具有常数量曲率的子流形,得到一个积分公式,利用这个积分公式证明了:欧氏空间中具常数量曲率的紧致超曲面必然是n维欧氏超球面的一个刚性.

Let x：Mn→Rn＋mbe a compact orientable Riemannian manifold isometrically immersed in the Euclidean space.For a compact orientable Riemannian manifold with constant scalar curvature in the Euclidean space Rn＋m,an integral formula is obtained.By using the formula,one gets a rigidity theorem that a hypersurface with constant scalar curvature in the Euclidean space Rn＋1 must be an n-dimensional hypersphere.