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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.
📌 3 Key Takeaways
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Objective & Hypothesis:Öz $\xi$-Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of $\lambda$-hypersurfaces.
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Key Findings & Data:Moreover, $\xi$-submanifolds are expected to take the place of submanifolds with parallel mean curvature vector.
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Conclusion & Impact:More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Anahtar Kelimeler $\xi$-Submanifold · Bernstein-type theorem
Özet
Öz $\xi$-Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of $\lambda$-hypersurfaces. Moreover, $\xi$-submanifolds are expected to take the place of submanifolds with parallel mean curvature vector. In this paper, we establish a Bernstein-type theorem for $\xi$-submanifolds in the Euclidean spaces. More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.
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Cite This Article
JİANG, X., SUN, H., ZHAO, P. (2019). A Bernstein-type theorem for $\xi$-submanifolds with flat normal bundle in the Euclidean spaces. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 1, 36–43. https://ajindex.com/articles/212743