论文标题
在欧几里得空间的爱因斯坦次级
On Einstein submanifolds of Euclidean space
论文作者
论文摘要
让扭曲的产品$ m^n = l^m \times_φf^{n-m} $，$ n \ geq m+3 \ geq 8 $，riemannian歧管的geq 8 $是爱因斯坦歧管，带有ricci曲率$ρ$，承认与euclidean space相连，并承认与euclidean space一起使用CORCIMESILY二。假设$ l^m $也是爱因斯坦，但不是恒定的截面曲率，这表明$ρ= 0 $，submanifold是本地的圆柱体，其尺寸至少为$ n-m $。因此，$ l^m $也是Ricci平坦的。如果$ m^n $完成，则如果$ l^m $的假设被其标量曲率$ s_l $是常数或$ s_l \ leq（2m-n）ρ$取代，则相同的结论在全球范围内。
Let the warped product $M^n=L^m\times_φF^{n-m}$, $n\geq m+3\geq 8$, of Riemannian manifolds be an Einstein manifold with Ricci curvature $ρ$ that admits an isometric immersion into Euclidean space with codimension two. Under the assumption that $L^m$ is also Einstein, but not of constant sectional curvature, it is shown that $ρ=0$ and that the submanifold is locally a cylinder with an Euclidean factor of dimension at least $n-m$. Hence $L^m$ is also Ricci flat. If $M^n$ is complete, then the same conclusion holds globally if the assumption on $L^m$ is replaced by the much weaker condition that either its scalar curvature $S_L$ is constant or that $S_L\leq (2m-n)ρ$.
