Tohoku Mathematical Journal
2025
June
SECOND SERIES VOL. 77, NO. 2

Tohoku Math. J.
77 (2025), 229-237

Title ON EINSTEIN SUBMANIFOLDS OF EUCLIDEAN SPACE

Author Marcos Dacjzer, Christos-Raent Onti and Theodoros Vlachos

(Received December 6, 2022)
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 a 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 $S_L\leq (2m-n)\rho$.

Mathematics Subject Classification. Primary 53C40; Secondary 53B25, 53C25.
Key words and phrases. Einstein submanifolds, scalar curvature, cylinders.

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