On Einstein submanifolds of Euclidean space

Let the warped product $M^n=L^m{\times }_{\phi }{F}^{n-m}$, $n\ge m+3\ge 8$, of Riemannian manifolds be an Einstein manifold with Ricci curvature $\rho$ 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 $\rho =0$ 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 $S_L\le (2m-n)\rho$.