On Einstein submanifolds of Euclidean space

TL;DR

This paper classifies Einstein warped product submanifolds in Euclidean space with codimension two, showing they are Ricci flat and locally cylindrical under certain conditions, extending understanding of their geometric structure.

Contribution

It proves that Einstein warped product submanifolds with specific conditions are Ricci flat and locally cylindrical, providing new classification results in submanifold geometry.

Findings

The Ricci curvature $ ho$ must be zero for such submanifolds.

The submanifold is locally a cylinder with an Euclidean factor.

Global results hold under weaker scalar curvature conditions.

Abstract

Let the warped product $M^n=L^m\times_\varphi F^{n-m}$, $n\geq m+3\geq 8$, 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 $S_L\leq (2m-n)\rho$.