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On Einstein submanifolds of Euclidean space
- Publication Year :
- 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 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 $S_L\leq (2m-n)\rho$.
- Subjects :
- Mathematics - Differential Geometry
53C25, 53C40, 53C42
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2210.09568
- Document Type :
- Working Paper